#497104

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂³	C₂×#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#97 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

426 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 80
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 80 points:

(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18,22,27,23,29,40,28)(19,24,33,45,31,43,59,46)(20,34,37,61,38,63,65,62)(21,39,35,26,41,51,64,52)(25,47,48,66,50,73,49,36)(30,53,57,75,55,74,79,76)(32,60,54,44,58,72,77,71)(42,67,69,78,68,80,70,56);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,10)(14,15)(17,19,22,33,23,31,40,59)(18,24,27,45,29,43,28,46)(20,30,37,57,38,55,65,79)(21,32,35,54,41,58,64,77)(25,42,48,69,50,68,49,70)(26,44,51,72,52,71,39,60)(34,53,61,75,63,74,62,76)(36,56,47,67,66,78,73,80);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,35,47,22,37,41,66,23,38,64,73,40,65,21,36)(18,25,39,62,28,49,52,63,29,50,51,61,27,48,26,34)(19,30,54,67,33,57,58,78,31,55,77,80,59,79,32,56)(24,42,60,76,46,70,71,74,43,68,72,75,45,69,44,53);(1,5,6,4)(2,8,9,14)(3,11,12,13)(7,16,10,15)(19,31)(24,43)(30,55)(32,58)(33,59)(42,68)(44,71)(45,46)(53,74)(54,77)(56,78)(57,79)(60,72)(67,80)(69,70)(75,76)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404300);