#512484

order	512
nilpotency class	4
isoclinism	isoclinic class of #511951
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×D₁₆	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

480 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 146 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)(3,8)(4,6)(5,7)(9,10)(11,13)(12,14)(15,16)(19,27)(20,28)(21,31)(23,33)(25,34)(26,38)(29,43)(30,41)(32,42)(35,44)(36,48)(37,47)(39,40)(45,46);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,33)(28,34)(29,36)(30,35)(31,38)(32,37)(39,45)(40,46)(41,44)(42,47)(43,48);(17,19,28,40,21,29,32,41,22,30,42,43,31,39,20,27)(18,23,34,46,26,36,37,44,24,35,47,48,38,45,25,33);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,24)(23,35)(25,37)(26,38)(33,44)(34,47)(36,45)(46,48)

Polycyclic presentation:

2^9|b^2=eg,c^2=fh,d^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,d^b=di

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419680);