#72299657

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁵
commutator	C₂⁴	C₂⁴×C₄
prime radical	C₂²×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

14693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 32 points:

(11,15)(14,16)(20,24)(22,23)(26,28)(30,31);(2,3)(5,6,8,7)(11,15)(14,16)(17,18,21,19)(20,22,24,23)(25,26,29,28)(27,30,32,31);(1,2,4,3)(10,13)(11,15)(18,19)(20,24)(26,28)(30,31);(2,3)(5,7,8,6)(9,10)(11,16)(12,13)(14,15)(17,19,21,18)(20,23,24,22)(25,27,29,32)(26,30,28,31);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,20,21,24)(18,22,19,23)(25,28,29,26)(27,31,32,30)

Polycyclic presentation:

2^10|b^2=fj,c^2=h,d^2=fj,e^2=hi,b^a=bf,c^b=cfg,d^c=dfhi,e^a=ei,e^b=efgi,e^c=ei,e^d=efgi

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72299663
maximal normal subgroup of	#72299663
maximal quotient of	#72299929

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