#6468665

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1574 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)(10,14)(15,16)(17,18,22,30,21,24,20,26)(19,25,28,23,27,31,29,32);(9,10,13,14)(11,16,12,15)(17,19,21,27)(18,23,24,32)(20,28,22,29)(25,26,31,30);(2,5)(6,8)(9,11,13,12)(10,15,14,16)(18,24)(19,27)(26,30)(28,29);(9,12,13,11)(10,16,14,15)(17,19)(18,25)(20,29)(21,27)(22,28)(23,30)(24,31)(26,32);(1,3,4,7)(2,6,5,8)(17,20,21,22)(18,26,24,30)(19,28,27,29)(23,31,32,25)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6375861);