#6884066

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682532
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#39131	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1509 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)(17,18,22,26)(19,28,25,24)(20,29,32,30)(21,27,23,31);(1,3)(2,5)(4,8)(6,7)(9,10)(11,15)(12,16)(13,14)(17,19,22,25)(18,24,26,28)(20,31,32,27)(21,29,23,30);(2,6)(3,8)(10,14)(15,16)(17,20,21,24,22,32,23,28)(18,25,31,30,26,19,27,29);(2,6)(5,7)(9,11,13,12)(10,15,14,16)(18,26)(19,25)(27,31)(29,30);(9,12,13,11)(10,15,14,16)(17,21,22,23)(18,27,26,31)(19,29,25,30)(20,28,32,24)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6791262);