#503523

order	512
nilpotency class	4
isoclinism	isoclinic class of #501060
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33179	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

825 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 825 conjugacy classes of subgroups — too many to draw. Only its 174 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:

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

Polycyclic presentation:

2^9|a^2=ehi,b^2=e,c^2=ef,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,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^c=gi

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, 410719);