#72323593

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹¹×C₄

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

33307 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^10|a^2=f,b^2=g,c^2=f,d^2=fghi,e^2=hi,b^a=bf,c^a=cfg,c^b=cj,d^a=dh,d^b=dfi,d^c=di,e^a=efh,e^b=efgi,e^c=egi,e^d=efg

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72323733, #72323737, #72323741
maximal normal subgroup of	#72323733, #72323737, #72323741

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