#508539

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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

508 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 132 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) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=fh,d^2=ei,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, 415735);