#499546

order	512
nilpotency class	4
isoclinism	isoclinic class of #499054
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

762 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 234 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) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406742);