#6999715

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1617 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6906911);