#302764

order	512
nilpotency class	3
isoclinism	isoclinic class of #268139
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

627 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 627 conjugacy classes of subgroups — too many to draw. Only its 156 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209960);