#6466619

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1430 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)(9,10,12,13)(11,15,16,14)(17,18,22,23)(19,20,26,28)(21,24,29,30)(25,27,31,32);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19,22,26)(18,20,23,28)(21,27,29,32)(24,31,30,25);(9,11,12,16)(10,14,13,15)(17,20,23,26,22,28,18,19)(21,27,30,31,29,32,24,25);(17,21,23,30,22,29,18,24)(19,25,20,27,26,31,28,32);(21,29)(24,30)(25,31)(27,32)

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=ghi,d^2=h,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,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, 6373815);