#293708

order	512
nilpotency class	3
isoclinism	isoclinic class of #271131
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

744 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 166 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)(11,14)(12,15)(13,16)(17,24)(18,22)(19,21)(20,23)(25,26,30,33)(27,31,34,39)(28,29,32,38)(35,40,36,37);(2,5)(6,7)(9,11,12,18)(10,14,15,22)(13,17,20,19)(16,24,23,21)(25,27,30,34)(26,31,33,39)(28,37,32,40)(29,35,38,36);(10,15)(11,17)(13,20)(14,21)(18,19)(22,24)(25,28,29,33,30,32,38,26)(27,31,35,37,34,39,36,40);(1,3,4,8)(2,6,5,7)(27,34)(31,39)(35,36)(37,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fhi,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,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, 200904);