#324464

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

658 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 231660);