#72332900

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 8638 conjugacy classes of subgroups; only the 1151 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^10|a^2=i,c^2=i,d^2=hj,b^a=bf,c^a=ci,c^b=cfh,d^a=dg,d^b=dfgh,d^c=dghij,e^a=efgij,e^b=eghj,e^c=ei,e^d=efh

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333232, #72333264, #72333330
maximal normal subgroup of	#72333232, #72333264, #72333330

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