#507666

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
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

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

normal subgroups only

This group has 590 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)(11,15)(12,13)(14,16)(17,18,23,36,34,43,33,42,22,27,21,26,20,25,38,48)(19,28,45,40,29,46,35,24,32,47,30,39,31,44,37,41);(1,3,4,8)(2,5,6,7)(10,13)(11,16);(9,11,12,16)(10,14,13,15)(17,19)(18,24)(20,29)(21,35)(22,32)(23,37)(25,39)(26,44)(27,41)(28,42)(30,33)(31,34)(36,46)(38,45)(40,43)(47,48);(2,6)(5,7)(10,13)(14,15)(17,20,22,34)(18,25,27,43)(19,29,32,31)(21,33,23,38)(24,39,41,40)(26,42,36,48)(28,46,47,44)(30,37,45,35)

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=de,c^b=cg,f^c=fh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414862);