#6700936

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×#447	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1474 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,13)(14,16)(19,27)(23,31)(28,32)(29,30);(10,13)(11,16)(17,18)(19,28)(20,24)(21,26)(22,25)(23,29)(27,32)(30,31);(2,5)(6,8)(9,11,12,16)(10,14,13,15)(17,19,21,27)(18,23,26,31)(20,29,22,30)(24,32,25,28);(1,3,4,7)(2,6,5,8)(10,13)(14,15)(17,20,21,22)(18,24,26,25)(19,29,27,30)(23,32,31,28);(18,25)(20,22)(23,28)(24,26)(29,30)(31,32)

Polycyclic presentation:

2^9|c^2=fi,d^2=fg,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, 6608132);