#508862

order	512
nilpotency class	4
isoclinism	isoclinic class of #508292
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

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

normal subgroups only

This group has 574 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)(17,18,22,29,20,27,38,48,21,28,23,30,24,31,39,42)(19,32,35,45,33,46,41,47,34,25,36,44,37,43,40,26);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,22,35,20,33,38,41,21,34,23,36,24,37,39,40)(18,25,29,44,27,43,48,26,28,32,30,45,31,46,42,47);(1,3)(2,6)(4,7)(5,8)(10,14)(13,16)(17,19,21,34)(18,26,28,47)(20,37,24,33)(22,40,23,41)(25,42,32,48)(27,44,31,45)(29,43,30,46)(35,38,36,39);(3,7)(6,8)(17,20,21,24)(18,27,28,31)(19,33,34,37)(22,38,23,39)(25,43,32,46)(26,45,47,44)(29,48,30,42)(35,41,36,40)

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,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, 416058);