#331490

order	512
nilpotency class	3
isoclinism	isoclinic class of #331276
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄³

Aliases

No aliases for this group.

Subgroups

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

719 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 719 conjugacy classes of subgroups — too many to draw. Only its 154 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,21)(18,22)(19,24)(20,23)(25,26,31,39,29,34,30,35)(27,36,38,40,28,32,37,33);(2,5)(6,8)(9,11,17,18,13,20,12,19)(10,14,21,22,15,23,16,24)(25,27,30,37,29,28,31,38)(26,32,35,40,34,36,39,33);(3,7)(6,8)(11,18)(12,17)(14,22)(16,21)(19,20)(23,24)(25,28)(26,33)(27,29)(30,37)(31,38)(32,39)(34,40)(35,36);(1,3,4,7)(2,6,5,8)(10,15)(14,23)(16,21)(22,24)(27,28)(32,36)(33,40)(37,38)

Polycyclic presentation:

2^9|a^2=fh,b^2=ghi,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,c^b=cg,d^b=dh,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, 238686);