#512184

order	512
nilpotency class	4
isoclinism	isoclinic class of #511887
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 611 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)(3,6)(4,5)(7,8)(9,10)(11,22)(12,27)(13,21)(14,20)(15,18)(16,23)(17,25)(19,24)(26,33)(28,37)(29,40)(30,32)(31,34)(35,38)(36,39);(2,5)(6,8)(9,11,13,25,16,26,14,15)(10,17,20,22,23,18,21,33)(12,24,28,38,30,39,29,31)(19,32,34,37,36,27,35,40);(1,3,4,7)(2,6,5,8)(9,11,13,25,16,26,14,15)(10,18,20,33,23,17,21,22)(12,24,28,38,30,39,29,31)(19,27,34,40,36,32,35,37);(9,12)(10,19)(11,24)(13,28)(14,29)(15,31)(16,30)(17,32)(18,27)(20,34)(21,35)(22,37)(23,36)(25,38)(26,39)(33,40)

Polycyclic presentation:

2^9|b^2=fi,c^2=e,e^2=i,f^2=i,g^2=hi,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=gh,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, 419380);