#330780

order	512
nilpotency class	3
isoclinism	isoclinic class of #329455
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×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

538 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 538 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=e,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, 237976);