#242963

order	512
nilpotency class	3
isoclinism	isoclinic class of #242962
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#397
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 692 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) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,f^b=fh,g^b=gi,d^c=di,f^c=fh,g^c=ghi

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, 150159);