#6435835

order	512
nilpotency class	3
isoclinism	isoclinic class of #6432861
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1446 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)(3,6)(4,5)(7,8)(11,15)(14,16)(17,18,23,28,22,24,21,27)(19,26,29,32,20,30,31,25);(2,5)(6,8)(9,10)(11,14)(12,13)(15,16)(18,24)(19,20)(27,28)(29,31);(10,13)(14,16)(17,19,22,20)(18,25,24,32)(21,29,23,31)(26,28,30,27);(9,11,12,15)(10,14,13,16)(17,20,22,19)(18,26,24,30)(21,29,23,31)(25,28,32,27);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,27,24,28)(19,29,20,31)(25,26,32,30)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343031);