#6595073

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1323 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)(3,7)(4,8)(5,6)(9,10,12,14)(11,13,15,16)(17,18)(19,23)(20,31)(21,25)(22,27)(24,35)(26,32)(28,42)(29,37)(30,38)(33,41)(34,40)(36,45)(39,43)(44,47)(46,48);(2,6)(7,8)(9,11)(10,13)(12,15)(14,16)(17,19,22,29)(18,23,27,37)(20,28,33,46)(21,30,32,43)(24,36,40,47)(25,38,26,39)(31,42,41,48)(34,44,35,45);(1,3,5,4)(2,7,6,8)(11,15)(13,16)(17,20,32,34,22,33,21,35)(18,24,25,41,27,40,26,31)(19,28,43,44,29,46,30,45)(23,36,38,48,37,47,39,42);(1,4,5,3)(2,8,6,7)(9,11)(10,13)(12,15)(14,16)(19,29)(20,33)(23,37)(24,40)(30,43)(31,41)(34,35)(38,39);(11,15)(13,16)(17,21,22,32)(18,25,27,26)(19,30,29,43)(20,34,33,35)(23,38,37,39)(24,31,40,41)(28,44,46,45)(36,42,47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fgi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^c=ei,e^d=eh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6502269);