#6757123

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667700
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2113 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)(11,14)(12,13)(15,16)(17,18,22,30,21,25,19,23)(20,27,26,24,29,31,28,32);(1,3)(2,5)(4,8)(6,7)(10,13)(11,15)(17,19,21,22)(18,23,25,30)(20,26,29,28)(24,31,32,27);(9,11,12,15)(10,14,13,16)(17,20,21,29)(18,24,25,32)(19,26,22,28)(23,31,30,27);(2,6)(5,7)(10,13)(14,16)(18,25)(23,30)(24,32)(27,31);(3,8)(5,7)(17,19,21,22)(18,23,25,30)(20,26,29,28)(24,31,32,27)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6664319);