#499662

order	512
nilpotency class	4
isoclinism	isoclinic class of #499662
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#23810
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #1747 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

614 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 614 conjugacy classes of subgroups — too many to draw. Only its 136 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=efh,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,d^b=dh,e^b=eh,f^b=fhi,h^b=hi,e^c=eh,f^c=fh,h^c=hi,g^d=gi

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, 406858);