#507394

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, prime radical	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 516 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) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=g,c^2=i,d^2=e,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, 414590);