#511958

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 970 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,e^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,h^a=hi,c^b=ci,d^c=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419154);