#6894256

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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	#37418	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1547 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6801452);