#512272

order	512
nilpotency class	4
isoclinism	isoclinic class of #512072
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

497 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 497 conjugacy classes of subgroups — too many to draw. Only its 138 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)(3,5)(4,6)(7,8)(9,10,13,11)(12,22,23,37)(14,28,26,39)(15,21,27,18)(16,32,19,24)(17,34,20,30)(25,35,31,38)(29,33,40,36);(1,3,4,7)(2,5,6,8)(9,11,13,10)(12,19,23,16)(14,17,26,20)(15,21,27,18)(22,32,37,24)(25,36,31,33)(28,34,39,30)(29,35,40,38);(2,6)(5,8)(10,11)(16,19)(17,20)(18,21)(22,37)(28,39)(33,36)(35,38);(9,12,15,31,13,23,27,25)(10,16,18,36,11,19,21,33)(14,29,30,32,26,40,34,24)(17,35,28,37,20,38,39,22)

Polycyclic presentation:

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

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, 419468);