#6463674

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437764
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	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1802 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)(4,5)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18,23,32,22,26,21,27)(19,28,29,24,25,31,30,20);(2,5)(6,7)(10,14)(11,12)(17,19)(18,24)(20,26)(21,29)(22,25)(23,30)(27,31)(28,32);(1,3,4,8)(2,6,5,7)(9,11,13,12)(10,15,14,16)(17,20,22,24)(18,25,26,19)(21,31,23,28)(27,29,32,30);(9,12,13,11)(10,15,14,16)(18,26)(19,25)(27,32)(29,30);(17,21,22,23)(18,27,26,32)(19,29,25,30)(20,28,24,31)

Polycyclic presentation:

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

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, 6370870);