#350033

order	512
nilpotency class	3
isoclinism	isoclinic class of #346882
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

861 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 861 conjugacy classes of subgroups — too many to draw. Only its 156 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)(9,10,13,16)(11,21,12,23)(14,20,15,19)(17,24,18,22)(25,26,30,28)(27,33,36,40)(29,37,39,32)(31,38,34,35);(2,5)(6,8)(10,16)(17,18)(19,20)(21,23);(1,3,4,7)(2,6,5,8)(9,11,13,12)(10,17,16,18)(14,24,15,22)(19,23,20,21)(25,27,30,36)(26,31,28,34)(29,38,39,35)(32,40,37,33);(9,12,13,11)(10,18,16,17)(14,24,15,22)(19,23,20,21)(25,28,30,26)(27,34,36,31)(29,32,39,37)(33,38,40,35)

Polycyclic presentation:

2^9|a^2=h,c^2=ei,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 257229);