#508879

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
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	C₂²×D₈	C₂³
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 523 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)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18,22,27,38,19,29,31,21,26,37,43,23,28,24,41)(20,32,35,44,40,48,47,42,34,46,45,30,36,33,39,25);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,31,38,26,29,43,21,28,37,41,23,18,24,27)(20,33,35,25,40,32,47,44,34,48,45,42,36,46,39,30);(3,7)(6,8)(10,14)(13,16)(17,20)(18,25)(19,30)(21,34)(22,39)(23,40)(24,35)(26,42)(27,33)(28,44)(29,45)(31,46)(32,41)(36,38)(37,47)(43,48);(1,3,4,7)(2,6,5,8)(20,34)(25,42)(30,44)(32,46)(33,48)(35,45)(36,40)(39,47)

Polycyclic presentation:

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

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, 416075);