#304718

order	512
nilpotency class	3
isoclinism	isoclinic class of #271387
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

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

744 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 178 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)(11,17)(12,14)(13,21)(15,19)(16,20)(18,23)(22,24)(25,26,31,36)(27,32,38,33)(28,39,37,40)(29,35,30,34);(1,3,5,4)(2,6,8,7)(10,14)(15,22)(16,21)(17,23)(25,27,31,38)(26,32,36,33)(28,29,37,30)(34,39,35,40);(1,3,5,4)(2,7,8,6)(9,11,13,19,12,18,20,24)(10,15,16,17,14,22,21,23)(25,28,30,27,31,37,29,38)(26,33,35,40,36,32,34,39);(1,4,5,3)(2,7,8,6)(10,14)(11,18)(16,21)(19,24)

Polycyclic presentation:

2^9|a^2=eh,b^2=eh,c^2=efh,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 211914);