#507672

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

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

normal subgroups only

This group has 584 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=eg,c^2=g,d^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, 414868);