#72317900

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#1946077
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#453952	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3876 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)(3,11)(4,8)(5,17)(6,10)(7,14)(9,19)(12,25)(13,26)(15,24)(16,23)(18,21)(20,28)(22,30)(27,32)(29,31);(1,3)(2,7,10,11)(4,12)(5,18,19,13)(6,14)(8,20,23,25)(9,21)(15,27,30,29)(16,28)(17,26)(22,32)(24,31);(1,4)(2,8)(3,12)(5,15)(6,16)(7,20)(9,24)(10,23)(11,25)(13,29)(14,28)(17,22)(18,27)(19,30)(21,31)(26,32);(1,5)(2,9)(3,13)(4,15)(6,19)(7,21)(8,22)(10,17)(11,26)(12,27)(14,18)(16,30)(20,31)(23,24)(25,32)(28,29);(7,11)(8,23)(21,26)(22,24)

Polycyclic presentation:

2^10|b^2=f,b^a=bh,c^a=cfi,d^a=dhj,d^b=dg,d^c=dhi,e^a=egi,e^b=ei,e^c=ei,f^a=fj,f^d=fj,g^a=gj,g^b=gj,g^c=gj,h^b=hj,i^a=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317935, #72318468, #72319027, …
maximal normal subgroup of	#72317935, #72318468, #72319027, …

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