#7024132

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996355
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1510 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)(17,18,22,26)(19,27,28,44)(20,24,35,43)(21,36,29,25)(23,37,38,31)(30,45,32,48)(33,42,34,41)(39,46,40,47);(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,14)(13,16)(19,28)(23,38)(27,44)(30,32)(31,37)(39,40)(45,48)(46,47);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,19,29,37,22,28,21,31)(18,23,36,44,26,38,25,27)(20,32,33,47,35,30,34,46)(24,40,41,45,43,39,42,48);(1,4,5,3)(2,7,6,8)(11,16)(13,15)(17,20)(18,24)(19,30)(21,34)(22,35)(23,39)(25,42)(26,43)(27,45)(28,32)(29,33)(31,47)(36,41)(37,46)(38,40)(44,48);(10,14)(13,15)(17,21,22,29)(18,25,26,36)(19,31,28,37)(20,33,35,34)(23,27,38,44)(24,41,43,42)(30,46,32,47)(39,48,40,45)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fgi,d^2=f,e^2=i,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^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, 6931328);