#6475496

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470374
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35252	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1857 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,d^c=di,e^c=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, 6382692);