#326816

order	512
nilpotency class	3
isoclinism	isoclinic class of #326783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1012 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1012 conjugacy classes of subgroups — too many to draw. Only its 160 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)(11,13)(12,20)(14,19)(15,17)(16,23)(18,24)(21,22)(25,26,29,32)(27,30,34,36)(28,31,35,38)(33,37,39,40);(2,5)(6,8)(9,11,13,10)(12,21,22,20)(14,19,15,17)(16,24,18,23)(25,27)(26,30)(28,33)(29,34)(31,37)(32,36)(35,39)(38,40);(1,3)(2,6)(4,7)(5,8)(9,12,13,22)(10,16,11,18)(14,24,15,23)(17,21,19,20)(30,36)(37,40);(3,7)(6,8)(10,11)(12,22)(17,19)(23,24)(25,28,29,35)(26,31,32,38)(27,33,34,39)(30,37,36,40)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,g^a=gi,c^b=cg,d^b=dh,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, 234012);