#6458374

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1666 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)(9,10,12,14)(11,15,16,13)(17,18,24,32,23,27,22,20)(19,28,29,25,30,31,21,26);(2,6)(7,8)(17,19,23,30)(18,25,27,26)(20,31,32,28)(21,22,29,24);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,13,14,15)(17,20,24,18,23,32,22,27)(19,26,29,28,30,25,21,31);(1,4,5,3)(2,8,6,7)(17,21,22,30,23,29,24,19)(18,26,20,31,27,25,32,28);(17,22,23,24)(18,20,27,32)(19,29,30,21)(25,31,26,28)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,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^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, 6365570);