#6485940

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470901
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1387 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,16)(13,14)(17,18,24,27)(19,28,31,21)(20,22,26,23)(25,29,32,30);(1,3,5,4)(2,6,8,7)(9,10)(11,16)(12,15)(13,14)(17,19,23,30,24,31,22,29)(18,25,26,28,27,32,20,21);(1,3,5,4)(2,7,8,6)(10,14)(15,16)(17,20,23,18,24,26,22,27)(19,21,30,25,31,28,29,32);(9,11,13,12)(10,15,14,16)(17,21,24,28)(18,19,27,31)(20,29,26,30)(22,32,23,25);(1,4,5,3)(2,6,8,7)(9,12,13,11)(10,15,14,16)(17,22,24,23)(18,26,27,20)(19,29,31,30)(21,25,28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,d^c=di,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, 6393136);