#72332983

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁵	C₂⁵
prime radical	C₂³×#120	C₂²
LCS	G ▷ C₂⁵ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 6477 conjugacy classes of subgroups; only the 1155 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)(21,22,24,26)(23,25,27,28);(1,2,3,4)(6,8)(9,10)(11,12)(13,14,17,16)(15,18,19,20)(21,23)(22,25)(24,27)(26,28);(1,2)(3,4)(15,19)(18,20)(21,23,24,27)(22,25,26,28);(2,4)(5,6,7,8)(13,15)(14,18)(16,20)(17,19)(21,23)(22,25)(24,27)(26,28);(1,2)(3,4)(5,6)(7,8)(13,16,17,14)(15,20,19,18)(21,22)(23,25)(24,26)(27,28)

Polycyclic presentation:

2^10|a^2=i,b^2=gij,c^2=i,d^2=hi,e^2=fgj,b^a=bg,c^b=cf,d^a=dgij,d^b=dfh,d^c=dgj,e^a=ei,e^b=efh,e^d=efh

Conjugacy classes & character table

154 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333234, #72333255, #72333336
maximal normal subgroup of	#72333234, #72333255, #72333336

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