#7039040

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

1319 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,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=eh,d^c=dhi,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, 6946236);