#6463731

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438017
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 ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1506 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)(9,10,13,15)(11,16,12,14)(17,18,23,24)(19,27,28,32)(20,25,31,30)(21,26,22,29);(11,12)(14,16)(17,19,22,30,23,28,21,25)(18,20,26,27,24,31,29,32);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(18,24)(19,28)(25,30)(26,29);(9,12,13,11)(10,14,15,16)(17,20,23,31)(18,25,24,30)(19,29,28,26)(21,32,22,27);(17,21,23,22)(18,26,24,29)(19,25,28,30)(20,27,31,32)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=gi,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,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, 6370927);