#415593

order	512
nilpotency class	3
isoclinism	isoclinic class of #393929
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1460 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 1460 conjugacy classes of subgroups — too many to draw. Only its 270 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=g,c^2=g,d^2=hi,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,f^b=fi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322789);