#72285192

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
prime radical	#453952	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

5925 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(5,6)(7,11)(8,18)(9,13)(10,14)(12,21)(15,28)(16,32)(17,27)(19,25)(20,29)(22,26)(23,33)(24,30)(31,36)(34,35);(1,2,3,4)(5,7)(6,11)(8,19)(9,22)(10,17)(12,28)(13,26)(14,27)(15,21)(16,24)(18,25)(20,31)(23,34)(29,36)(30,32)(33,35);(12,18)(13,30)(15,19)(16,22);(5,8)(6,12)(7,15,17,19)(9,23)(10,21)(11,25,27,28)(13,29)(14,18)(16,31,22,34)(20,24)(26,36,32,35)(30,33);(5,9)(6,13,14,30)(7,16)(8,20)(10,24)(11,26,27,32)(12,29,18,33)(15,31)(17,22)(19,34)(21,23)(25,35,28,36)

Polycyclic presentation:

2^10|b^2=f,d^2=h,e^2=hi,b^a=bf,c^a=cg,c^b=cg,d^a=dj,d^b=dg,d^c=di,e^a=eghj,e^b=ehij,e^c=ei,e^d=eij,g^d=gj,g^e=gj,h^b=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285318, #72285424, #72285469
maximal normal subgroup of	#72285318, #72285424, #72285469

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