#6692596

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667604
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 ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1581 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)(4,6)(5,8)(9,10,13,15)(11,14,12,16)(17,18,21,22)(19,23,27,26)(20,28,30,24)(25,29,32,31);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,15,16)(17,19,21,27)(18,23,22,26)(20,29,30,31)(24,25,28,32);(9,12,13,11)(10,14,15,16)(17,20,21,30)(18,24,22,28)(19,25,27,32)(23,31,26,29);(2,6)(5,7)(17,18,21,22)(19,26,27,23)(20,24,30,28)(25,29,32,31);(3,8)(5,7)(18,22)(19,26)(23,27)(24,28)(25,29)(31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6599792);