#507689

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, 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

514 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 514 conjugacy classes of subgroups — too many to draw. Only its 162 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)(11,15)(13,16)(17,18,23,34,33,42,32,25,22,28,20,26,21,27,36,41)(19,24,35,45,44,48,43,37,31,40,29,38,30,39,46,47);(2,5)(6,7)(9,10,12,14)(11,16,15,13)(17,19,22,31)(18,24,28,40)(20,29,23,35)(21,30,33,44)(25,37,41,47)(26,38,34,45)(27,39,42,48)(32,43,36,46);(9,11,12,15)(10,13,14,16)(18,25)(20,32)(21,33)(23,36)(24,37)(26,27)(28,41)(29,43)(30,44)(34,42)(35,46)(38,39)(40,47)(45,48);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=fgi,b^2=gi,c^2=g,d^2=e,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, 414885);