#508515

order	512
nilpotency class	4
isoclinism	isoclinic class of #508494
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

479 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 479 conjugacy classes of subgroups — too many to draw. Only its 134 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)(9,10,13,12)(11,19,21,16)(14,24,26,17)(15,18,27,25)(20,32,38,28)(22,33,39,37)(23,36,31,34)(29,30,40,35);(2,5)(6,8)(9,11,15,31,13,21,27,23)(10,16,18,36,12,19,25,34)(14,28,30,22,26,32,35,39)(17,20,29,33,24,38,40,37);(1,3)(2,6)(4,7)(5,8)(9,12,13,10)(11,20,21,38)(14,29,26,40)(15,18,27,25)(16,32,19,28)(17,35,24,30)(22,34,39,36)(23,33,31,37);(3,7)(6,8)(10,12)(16,19)(17,24)(18,25)(20,38)(29,40)(33,37)(34,36)

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,d^c=de,g^c=ghi,h^c=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415711);