#72306949

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#13086
commutator	C₂²×D₄	C₂³×C₄
Frattini, cosocle	C₂×#127	C₂⁴
prime radical	#6437456	C₂
LCS	G ▷ C₂²×D₄ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁴ ◁ C₂²×#41 ◁ C₂×#3498 ◁ G
LECS	G ▷ C₂×#127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

4307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^10|a^2=g,d^2=j,g^2=ij,h^2=j,b^a=be,c^a=cei,c^b=cj,d^a=def,d^b=de,d^c=de,e^a=ej,f^a=fh,f^b=fj,g^b=gj,g^c=gj,g^d=gh,g^f=gij,h^a=hi,h^d=hj,h^f=hj,i^d=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306956, #72306961, #72307002, …
maximal normal subgroup of	#72306956, #72306961, #72307002, …

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, 1, F.7);;
gap> SetPower(coll, 4, F.10);;
gap> SetPower(coll, 7, F.9*F.10);;
gap> SetPower(coll, 8, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.5);;
gap> SetConjugate(coll, 3, 1, F.3*F.5*F.9);;
gap> SetConjugate(coll, 3, 2, F.3*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.5*F.6);;
gap> SetConjugate(coll, 4, 2, F.4*F.5);;
gap> SetConjugate(coll, 4, 3, F.4*F.5);;
gap> SetConjugate(coll, 5, 1, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
gap> SetConjugate(coll, 7, 4, F.7*F.8);;
gap> SetConjugate(coll, 7, 6, F.7*F.9*F.10);;
gap> SetConjugate(coll, 8, 1, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.10);;
gap> SetConjugate(coll, 8, 6, F.8*F.10);;
gap> SetConjugate(coll, 9, 4, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);