#326700

order	512
nilpotency class	3
isoclinism	isoclinic class of #326683
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, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1246 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1246 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,18)(11,15,21,24)(12,16,13,17)(19,22,20,23)(25,26)(27,30)(28,31)(29,32)(33,38)(34,37)(35,39)(36,40);(2,5)(6,8)(9,11)(10,15)(12,20)(13,19)(14,21)(16,23)(17,22)(18,24)(25,27)(26,30)(28,33)(29,34)(31,38)(32,37)(35,40)(36,39);(1,3)(2,6)(4,7)(5,8)(10,16)(11,19)(12,13)(17,18)(20,21)(22,23)(25,28,29,35)(26,31,32,39)(27,33,34,40)(30,36,37,38);(3,7)(6,8)

Polycyclic presentation:

2^9|a^2=h,c^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=de,f^c=fh,g^c=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 233896);