#72332872

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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 8404 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)(9,10)(11,12)(13,14,17,18)(15,16,19,20)(21,22,25,28)(23,24,27,26);(1,2)(3,4)(10,11)(21,23,25,27)(22,26,28,24);(15,19)(16,20)(23,27)(24,26);(2,4)(5,6,7,8)(9,11,12,10)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28);(1,2)(3,4)(5,6)(7,8)(13,16)(14,15)(17,20)(18,19)(21,24)(22,27)(23,28)(25,26)

Polycyclic presentation:

2^10|a^2=i,b^2=fij,d^2=h,b^a=bf,c^b=cfij,d^a=dg,d^b=dfgij,d^c=di,e^a=efj,e^b=efij,e^c=ei,e^d=efghij

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333231, #72333258, #72333330
maximal normal subgroup of	#72333231, #72333258, #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, 2, F.6*F.9*F.10);;
gap> SetPower(coll, 4, F.8);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.6*F.9*F.10);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.9);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);