#6856422

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667700
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	#2856	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1677 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)(9,10,13,12)(11,14,16,15)(17,18)(19,26)(20,28)(21,25)(22,23)(24,27)(29,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,11,13,16)(10,14,12,15)(17,19,21,22)(18,23,25,26)(20,27,29,30)(24,28,31,32);(3,8)(5,7)(9,10,13,12)(11,15,16,14)(17,20,21,29)(18,24,25,31)(19,27,22,30)(23,28,26,32);(2,6)(5,7)(18,25)(23,26)(24,31)(28,32);(9,12,13,10)(11,14,16,15)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=fh,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6763618);