#72298293

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁵	C₂⁵
commutator	C₂⁴	C₂⁴×C₄
prime radical	C₂²×#2669	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₁

At least 7878 conjugacy classes of subgroups; only the 1657 normal subgroups are shown.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^10|b^2=fh,d^2=gh,e^2=hj,b^a=bf,c^a=cfg,c^b=cfg,d^a=dhi,d^b=dfg,d^c=dfg,e^a=efg,e^b=efj,e^c=ehi,e^d=efj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72298577, #72298598, #72298605
maximal normal subgroup of	#72298577, #72298598, #72298605

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