#305335

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

620 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 620 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 212531);