#330364

order	512
nilpotency class	3
isoclinism	isoclinic class of #329743
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

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

normal subgroups only

This group has 553 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)(3,6)(4,5)(7,8)(9,10,15,23,14,17,13,18)(11,16,24,21,19,12,20,22)(25,26,30,32,29,34,28,33)(27,31,37,40,36,38,35,39);(2,5)(6,8)(9,11)(10,16)(12,17)(13,20)(14,19)(15,24)(18,22)(21,23)(25,27,29,36)(26,31,34,38)(28,35,30,37)(32,40,33,39);(3,7)(6,8)(9,12,14,16)(10,11,17,19)(13,22,15,21)(18,24,23,20)(26,32)(27,35)(28,30)(33,34)(36,37)(39,40);(1,3,4,7)(2,6,5,8)(10,17)(11,19)(18,23)(20,24)(27,36)(31,38)(35,37)(39,40)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 237560);