#503249

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2215	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1112 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 1112 conjugacy classes of subgroups — too many to draw. Only its 194 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=fi,d^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410445);