#6567443

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535823
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1336 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)(9,10,14,13)(11,16,12,15)(18,23)(19,26)(21,30)(24,29)(25,31)(27,32);(1,3,5,4)(2,6,8,7)(9,11,14,12)(10,15,13,16)(17,18,21,23,22,25,30,31)(19,24,26,20,27,29,32,28);(1,3,5,4)(2,7,8,6)(9,10,14,13)(11,16,12,15)(17,19,21,26,22,27,30,32)(18,24,23,20,25,29,31,28);(1,3,5,4)(2,7,8,6)(9,12,14,11)(10,15,13,16)(17,20)(18,19)(21,29)(22,28)(23,26)(24,30)(25,27)(31,32);(1,4,5,3)(2,6,8,7)(9,13,14,10)(11,16,12,15)(19,27)(20,28)(24,29)(26,32)

Polycyclic presentation:

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

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, 6474639);