#491120

order	512
nilpotency class	4
isoclinism	isoclinic class of #491120
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

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

normal subgroups only

This group has 392 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)(6,25)(11,39)(12,43)(13,31)(15,47)(16,17)(18,45)(19,46)(21,55)(22,23)(24,54)(26,33)(27,35)(28,42)(29,30)(34,60)(36,37)(41,53)(44,51)(48,62)(49,50)(52,56)(58,59)(61,63);(1,2,9,30,6,12,26,31,7,13,33,43,25,29,5,8)(3,10,14,35,17,36,45,59,19,38,46,58,18,37,16,27)(4,21,32,55,20,53,23,28,24,11,40,39,54,42,22,41)(15,48,44,61,50,60,56,57,52,34,49,63,51,62,47,64);(1,3)(2,10)(4,15)(5,16)(6,17)(7,19)(8,27)(9,14)(11,34)(12,36)(13,38)(18,25)(20,50)(21,48)(22,47)(23,56)(24,52)(26,45)(28,57)(29,37)(30,35)(31,59)(32,44)(33,46)(39,63)(40,49)(41,64)(42,62)(43,58)(51,54)(53,60)(55,61);(1,4,6,20,7,24,25,54)(2,11,12,42,13,21,29,53)(3,15,17,50,19,52,18,51)(5,22,9,32,26,23,33,40)(8,28,30,39,31,41,43,55)(10,34,36,62,38,48,37,60)(14,44,45,56,46,49,16,47)(27,57,35,63,59,64,58,61)

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,d^b=di,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, 398316);