#505676

order	512
nilpotency class	4
isoclinism	isoclinic class of #500676
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

577 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 577 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,d^2=ei,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412872);