#468900

order	512
nilpotency class	3
isoclinism	isoclinic class of #454045
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1196 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1196 conjugacy classes of subgroups — too many to draw. Only its 190 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=efg,b^2=egi,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^c=de,h^c=hi,f^d=fi

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, 376096);