#6458443

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
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	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1670 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

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, 6365639);