#513177

order	512
nilpotency class	5
isoclinism	isoclinic class of #513173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	#33846	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

439 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|b^2=i,c^2=i,e^2=gh,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dgh,e^a=eg,g^a=gh,h^a=hi,c^b=ci,d^b=de,e^b=eg,g^b=gh,h^b=hi,e^d=eg,g^d=gh,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420373);