#508874

order	512
nilpotency class	4
isoclinism	isoclinic class of #508304
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

562 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 562 conjugacy classes of subgroups — too many to draw. Only its 150 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)(17,18,22,29,21,37,38,48,25,31,23,30,24,28,39,42)(19,26,33,44,36,43,40,27,20,35,34,45,32,46,41,47);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,33,21,36,38,40,25,20,23,34,24,32,39,41)(18,26,29,44,37,43,48,27,31,35,30,45,28,46,42,47);(1,3)(2,6)(4,7)(5,8)(10,14)(13,16)(17,20,25,19)(18,27,31,47)(21,36,24,32)(22,40,23,41)(26,42,35,48)(28,45,37,44)(29,43,30,46)(33,39,34,38);(3,7)(6,8)(17,21,25,24)(18,28,31,37)(19,32,20,36)(22,38,23,39)(26,43,35,46)(27,45,47,44)(29,42,30,48)(33,41,34,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=f,c^2=i,d^2=i,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, 416070);