#7001188

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1575 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,4)(3,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,18,23,29)(19,24,34,42)(20,26,36,45)(21,28,22,27)(25,37,44,35)(30,46,31,39)(32,41,33,40)(38,48,43,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,31)(21,32)(22,33)(23,34)(25,43)(26,39)(27,40)(28,41)(29,42)(30,36)(35,48)(37,47)(38,44)(45,46);(9,11)(10,13)(12,16)(14,15)(17,20,23,36)(18,25,29,44)(19,30,34,31)(21,35,22,37)(24,38,42,43)(26,28,45,27)(32,47,33,48)(39,40,46,41);(1,2,5,4)(3,7,8,6)(11,16)(13,15)(17,20)(18,26)(19,31)(21,37)(22,35)(23,36)(24,39)(25,27)(28,44)(29,45)(30,34)(32,47)(33,48)(38,41)(40,43)(42,46);(1,4,5,2)(3,7,8,6)(10,14)(13,15)(17,21,23,22)(18,27,29,28)(19,32,34,33)(20,35,36,37)(24,40,42,41)(25,26,44,45)(30,47,31,48)(38,46,43,39)

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6908384);