#325958

order	512
nilpotency class	3
isoclinism	isoclinic class of #289819
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

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

normal subgroups only

This group has 492 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)(11,19,20,17)(12,15,21,24)(13,22,23,18)(25,26,28,31)(27,33,35,30)(29,32,37,40)(34,39,36,38);(9,11,13,12,14,20,23,21)(10,15,18,19,16,24,22,17)(25,27,29,36,28,35,37,34)(26,30,32,39,31,33,40,38);(1,3,4,8)(2,5,6,7)(10,16)(15,24)(17,19)(18,22)(27,34)(29,37)(30,38)(32,40)(33,39)(35,36);(9,12,13,20,14,21,23,11)(10,17,18,15,16,19,22,24)(26,31)(27,35)(32,40)(34,36)

Polycyclic presentation:

2^9|a^2=fi,b^2=egi,c^2=fh,d^2=ei,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dhi,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, 233154);