#326953

order	512
nilpotency class	3
isoclinism	isoclinic class of #326925
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	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

934 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 934 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 234149);