#6701985

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1353 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)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,18,22,23)(19,26,29,24)(20,30,21,25)(27,32,31,28);(9,11,14,10)(12,15,13,16)(17,19)(18,24)(20,27)(21,31)(22,29)(23,26)(25,32)(28,30);(2,5)(6,8)(9,12,14,13)(10,15,11,16)(18,23)(19,26)(24,29)(25,30)(27,28)(31,32);(1,3,4,7)(2,6,5,8)(9,13,14,12)(10,15,11,16)(17,20)(18,25)(19,27)(21,22)(23,30)(24,32)(26,28)(29,31);(17,21,22,20)(18,25,23,30)(19,28,29,32)(24,31,26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=fg,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,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, 6609181);