#506383

order	512
nilpotency class	4
isoclinism	isoclinic class of #502469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

610 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|b^2=egh,d^2=ehi,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413579);