#491117

order	512
nilpotency class	4
isoclinism	isoclinic class of #491117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#19258
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

393 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 393 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|b^2=eg,d^2=g,e^2=gi,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,g^a=gi,h^a=hi,f^c=fh,h^c=hi,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398313);