#467072

order	512
nilpotency class	3
isoclinism	isoclinic class of #454045
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1518 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1518 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=egi,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, 374268);