#293846

order	512
nilpotency class	3
isoclinism	isoclinic class of #271771
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

717 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 717 conjugacy classes of subgroups — too many to draw. Only its 150 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)(3,7)(4,5)(6,8)(9,10,14,18)(11,15,16,24)(12,21,22,20)(13,19,23,17)(25,26,29,33)(27,34,37,31)(28,38,30,32)(35,40,36,39);(2,5)(6,7)(9,11)(10,15)(12,19)(13,20)(14,16)(17,22)(18,24)(21,23)(25,27,29,37)(26,31,33,34)(28,36,30,35)(32,39,38,40);(9,12,13,15,14,22,23,24)(10,16,17,21,18,11,19,20)(27,35)(28,30)(31,39)(32,38)(34,40)(36,37);(1,3,4,8)(2,6,5,7)(11,16)(12,22)(15,24)(20,21)(25,28,29,30)(26,32,33,38)(27,36,37,35)(31,39,34,40)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=f,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^b=gi,d^c=dhi,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, 201042);