#6468020

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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1346 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)(11,14)(12,13)(15,16)(18,24)(19,28)(26,30)(27,32);(1,3,4,8)(2,5,6,7)(17,18,22,24)(19,29,28,20)(21,27,23,32)(25,26,31,30);(10,13)(14,16)(17,19,23,30,22,28,21,26)(18,25,27,29,24,31,32,20);(17,20,23,25,22,29,21,31)(18,26,27,19,24,30,32,28);(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,27,24,32)(19,26,28,30)(20,25,29,31)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=h,d^2=h,e^2=g,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,d^c=di,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, 6375216);