#491119

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
prime radical	#33692	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #405 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 387 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)(5,24)(6,25)(9,26)(11,40)(12,32)(13,31)(15,46)(16,17)(18,45)(19,33)(21,22)(23,52)(27,36)(28,42)(29,30)(35,59)(37,38)(41,57)(43,58)(44,49)(47,48)(50,53)(55,56)(60,63)(61,62);(1,2,9,32,6,12,5,8,7,13,24,30,25,29,26,31)(3,10,33,55,17,37,16,27,19,39,14,36,18,38,45,56)(4,11,34,58,20,42,21,28,23,43,51,40,52,57,22,41)(15,35,47,60,48,61,46,54,50,62,44,63,49,59,53,64);(1,3)(2,10)(4,15)(5,16)(6,17)(7,19)(8,27)(9,33)(11,35)(12,37)(13,39)(14,24)(18,25)(20,48)(21,46)(22,53)(23,50)(26,45)(28,54)(29,38)(30,36)(31,56)(32,55)(34,47)(40,63)(41,64)(42,61)(43,62)(44,51)(49,52)(57,59)(58,60);(1,4,6,20,7,23,25,52)(2,11,12,42,13,43,29,57)(3,15,17,48,19,50,18,49)(5,21,24,51,26,22,9,34)(8,28,30,40,31,41,32,58)(10,35,37,61,39,62,38,59)(14,44,45,53,33,47,16,46)(27,54,36,63,56,64,55,60)

Polycyclic presentation:

2^9|b^2=egi,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, 398315);