#316163

order	512
nilpotency class	3
isoclinism	isoclinic class of #316149
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

501 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 501 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=egi,b^2=e,c^2=efi,d^2=eh,e^2=h,f^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^c=gh

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, 223359);