#507373

order	512
nilpotency class	4
isoclinism	isoclinic class of #501046
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

662 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,23,36,33,41,34,43,21,25,20,26,22,27,38,47)(19,28,45,40,35,48,32,24,29,46,30,44,31,42,37,39);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(19,29)(24,39)(28,46)(30,45)(31,35)(32,37)(40,44)(42,48);(11,15)(13,16)(17,19,21,29)(18,24,25,39)(20,32,23,37)(22,35,33,31)(26,42,36,48)(27,44,41,40)(28,47,46,43)(30,38,45,34);(2,6)(5,7)(18,25)(24,39)(26,36)(27,41)(28,46)(40,44)(42,48)(43,47)

Polycyclic presentation:

2^9|a^2=fg,b^2=egi,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414569);