#508897

order	512
nilpotency class	4
isoclinism	isoclinic class of #508304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 632 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:

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

Polycyclic presentation:

2^9|a^2=egi,b^2=f,d^2=e,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=di,d^c=de,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416093);