#353103

order	512
nilpotency class	3
isoclinism	isoclinic class of #352093
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

606 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 606 conjugacy classes of subgroups — too many to draw. Only its 218 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)(11,17)(12,21)(13,15)(14,20)(16,19)(18,23)(22,24)(25,26)(27,33)(28,30)(29,37)(31,35)(32,36)(34,39)(38,40);(1,3,4,8)(2,5,6,7)(11,18)(14,22)(17,23)(20,24)(27,34)(31,38)(33,39)(35,40);(9,11,19,24,13,18,12,20)(10,14,21,23,15,22,16,17)(26,30)(31,38)(32,37)(33,39);(10,15)(14,22)(16,21)(17,23)(25,27,29,35,28,34,36,40)(26,31,32,33,30,38,37,39)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=fhi,d^2=gh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=gi,c^b=ch,d^b=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260299);