#6488762

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469995
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1639 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)(9,10)(11,14)(12,15)(13,16)(17,18,22,27,23,26,20,25)(19,28,30,21,31,32,29,24);(11,12)(14,15)(17,19,23,31)(18,24,26,21)(20,30,22,29)(25,28,27,32);(2,5)(6,8)(9,11,13,12)(10,14,16,15)(17,20,23,22)(18,25,26,27)(19,29,31,30)(21,28,24,32);(9,12,13,11)(10,15,16,14)(17,21,23,24)(18,19,26,31)(20,28,22,32)(25,29,27,30);(1,3,4,7)(2,6,5,8)(10,16)(14,15)(18,26)(21,24)(25,27)(28,32)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,f^b=fi,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, 6395958);