#72331624

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷×C₄³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁵
commutator	C₂⁴	C₂⁴×C₄
prime radical	C₂×#15258	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 7588 conjugacy classes of subgroups; only the 1353 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:

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

Polycyclic presentation:

2^10|a^2=j,d^2=i,e^2=hi,b^a=bf,d^a=dh,d^b=di,d^c=dg,e^a=eh,e^b=ehi,e^c=ef

Conjugacy classes & character table

148 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72331633, #72331636, #72331641
maximal normal subgroup of	#72331633, #72331636, #72331641

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