#331789

order	512
nilpotency class	3
isoclinism	isoclinic class of #331655
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

624 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 142 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)(3,7)(4,6)(5,8)(9,10,15,19)(11,16,14,24)(12,22,13,23)(17,21,18,20)(25,26,30,29)(27,31,34,37)(28,32,36,38)(33,39,40,35);(1,3)(2,5)(4,8)(6,7)(9,11,15,14)(10,16,19,24)(12,20,13,21)(17,23,18,22)(25,27,30,34)(26,31,29,37)(28,35,36,39)(32,33,38,40);(9,12,15,13)(10,17,19,18)(11,20,14,21)(16,23,24,22)(25,28,30,36)(26,32,29,38)(27,33,34,40)(31,39,37,35);(2,6)(3,8)(9,13,15,12)(10,18,19,17)(11,21,14,20)(16,22,24,23)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 238985);