#330200

order	512
nilpotency class	3
isoclinism	isoclinic class of #329455
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

574 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,24)(18,22)(19,21)(20,23)(25,26,30,34,31,28,29,33)(27,32,39,40,36,38,37,35);(2,5)(6,8)(9,11,12,18)(10,14,15,22)(13,17,20,19)(16,24,23,21)(25,27,31,36)(26,32,28,38)(29,37,30,39)(33,35,34,40);(3,7)(6,8)(10,15)(11,17)(13,20)(14,21)(18,19)(22,24)(25,28,31,26)(27,35,36,40)(29,33,30,34)(32,39,38,37);(1,3,4,7)(2,6,5,8)(27,36)(32,38)(35,40)(37,39)

Polycyclic presentation:

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

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, 237396);