#6466618

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fg,c^2=gh,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, 6373814);