#72264702

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂³×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁵
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₄×#392 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5305 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=fg,d^2=h,e^2=i,i^2=j,c^a=cf,c^b=cfg,d^a=dgh,d^c=dgj,e^a=ej,e^b=ei,e^c=egh,e^d=ej,i^b=ij

Conjugacy classes & character table

106 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72264843, #72265530
maximal normal subgroup of	#72264843, #72265530

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