#497090

order	512
nilpotency class	3
isoclinism	isoclinic class of #3198
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂³	C₂×#131
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

432 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 218 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) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ef,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404286);