#6458449

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438991
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1528 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18,22,26,23,27,21,25)(19,28,20,31,30,29,32,24);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,21)(22,32)(23,30)(25,28)(26,29)(27,31);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,21,19,23,32,22,30)(18,24,25,29,27,31,26,28);(1,4,5,3)(2,8,6,7)(17,18,22,26,23,27,21,25)(19,29,20,24,30,28,32,31);(17,21,23,22)(18,25,27,26)(19,20,30,32)(24,28,31,29)

Polycyclic presentation:

2^9|a^2=h,b^2=gi,c^2=fg,d^2=f,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365645);