#72231851

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×B(4,𝔽₂)
commutator, Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁵
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

8021 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,15)(4,10)(5,23)(6,12)(7,9)(8,14)(11,21)(13,25)(16,28)(17,30)(18,32)(19,31)(20,26)(22,27)(24,29);(2,9)(10,26)(11,27)(12,13)(14,28)(15,31)(23,29)(30,32);(1,3,16,4)(2,10,28,15)(5,17,6,21)(7,19,8,20)(9,26,14,31)(11,12,30,23)(13,32,29,27)(18,25,22,24);(1,3,16,4)(2,11,28,30)(5,21,6,17)(7,19,8,20)(9,27,14,32)(10,12,15,23)(13,31,29,26)(18,24,22,25);(1,4,16,3)(2,11,9,27)(5,22,6,18)(7,20,8,19)(10,29,26,23)(12,15,13,31)(14,32,28,30)(17,24,21,25)

Polycyclic presentation:

2^10|c^2=h,d^2=h,e^2=hij,f^2=h,g^2=hj,b^a=bhj,c^a=ch,d^a=df,e^a=eg,e^c=eij,e^d=eij,f^a=fh,f^d=fh,f^e=fhi,g^a=ghj,g^c=gj,g^d=ghi,g^e=gh,g^f=gj,i^a=ij

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231852, #72231951, #72234741, …
maximal normal subgroup of	#72231951, #72234741, #72234789, …
maximal quotient of	#72237745, #72237869, #72284543, …
automorphism group of	#7394, #7395, #7426, …

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 3, F.8);;
gap> SetPower(coll, 4, F.8);;
gap> SetPower(coll, 5, F.8*F.9*F.10);;
gap> SetPower(coll, 6, F.8);;
gap> SetPower(coll, 7, F.8*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.8*F.10);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 4, 1, F.4*F.6);;
gap> SetConjugate(coll, 5, 1, F.5*F.7);;
gap> SetConjugate(coll, 5, 3, F.5*F.9*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.9*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 4, F.6*F.8);;
gap> SetConjugate(coll, 6, 5, F.6*F.8*F.9);;
gap> SetConjugate(coll, 7, 1, F.7*F.8*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8*F.9);;
gap> SetConjugate(coll, 7, 5, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);