#7030596

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1438 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)(11,13)(12,14)(15,16)(17,18,23,28)(19,29,30,41)(20,25,34,40)(21,27,22,26)(24,35,37,32)(31,43,44,48)(33,42,36,39)(38,45,47,46);(1,2,5,6)(3,8,4,7)(9,10)(11,15)(12,14)(13,16)(19,30)(24,37)(29,41)(31,44)(32,35)(38,47)(43,48)(45,46);(9,11)(10,13)(12,16)(14,15)(17,19,21,35,23,30,22,32)(18,24,26,41,28,37,27,29)(20,31,36,45,34,44,33,46)(25,38,42,48,40,47,39,43);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,20)(18,25)(19,31)(21,36)(22,33)(23,34)(24,38)(26,42)(27,39)(28,40)(29,43)(30,44)(32,46)(35,45)(37,47)(41,48);(1,4,5,3)(2,7,6,8)(10,14)(13,15)(17,21,23,22)(18,26,28,27)(19,32,30,35)(20,33,34,36)(24,29,37,41)(25,39,40,42)(31,45,44,46)(38,48,47,43)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gi,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=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6937792);