#511906

order	512
nilpotency class	4
isoclinism	isoclinic class of #511905
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#32902
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1056 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=ei,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,g^a=gi,h^a=hi,f^c=fh,h^c=hi,g^d=gi

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, 419102);