#325915

order	512
nilpotency class	3
isoclinism	isoclinic class of #288899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

500 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 170 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)(9,10,14,16)(11,17,19,24)(12,21,20,15)(13,23,22,18)(25,26)(27,30)(28,32)(29,31)(33,37)(34,39)(35,38)(36,40);(9,11,13,20,14,19,22,12)(10,15,18,17,16,21,23,24)(25,27,28,34,29,35,36,33)(26,30,32,39,31,38,40,37);(1,3,4,8)(2,5,6,7)(10,16)(11,19)(12,20)(18,23)(27,33)(28,36)(30,37)(32,40)(34,35)(38,39);(9,12,22,19,14,20,13,11)(10,17,23,15,16,24,18,21)(26,31)(30,38)(32,40)(37,39)

Polycyclic presentation:

2^9|a^2=f,b^2=egh,c^2=fh,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^c=dh,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, 233111);