#6691069

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667508
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	C₂×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 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)(3,6)(4,8)(5,7)(9,10)(11,13)(12,15)(14,16)(19,21)(24,26)(28,31)(29,30);(1,3,5,4)(2,6,7,8)(9,11,13,10)(12,15,16,14)(17,18,22,27)(19,28,21,31)(20,25,23,32)(24,30,26,29);(1,4,5,3)(2,6,7,8)(10,11)(14,15)(17,19,22,21)(18,24,27,26)(20,29,23,30)(25,31,32,28);(9,12,13,16)(10,14,11,15)(17,20,22,23)(18,25,27,32)(19,29,21,30)(24,31,26,28);(17,21,22,19)(18,26,27,24)(20,30,23,29)(25,28,32,31)

Polycyclic presentation:

2^9|b^2=g,c^2=fg,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6598265);