#512231

order	512
nilpotency class	4
isoclinism	isoclinic class of #511998
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#135	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

492 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=fi,c^2=e,e^2=i,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,g^a=gh,h^a=hi,g^d=ghi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419427);