#6458453

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
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

1514 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)(17,18,23,26,24,27,22,21)(19,28,29,32,30,31,20,25);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,24,30)(18,25,27,32)(20,22,29,23)(21,28,26,31);(1,3,5,4)(2,7,6,8)(9,12,13,11)(10,14,15,16)(17,20,23,19,24,29,22,30)(18,25,26,28,27,32,21,31);(1,4,5,3)(2,8,6,7)(17,21,23,18,24,26,22,27)(19,25,29,28,30,32,20,31);(17,22,24,23)(18,21,27,26)(19,29,30,20)(25,28,32,31)

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,d^a=di,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365649);