#6894259

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	#29602	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1598 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,d^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, 6801455);