#418486

order	512
nilpotency class	3
isoclinism	isoclinic class of #418484
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1624 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)(5,6,12,18)(7,19,23,38)(8,14,28,42)(9,29,32,15)(10,17,11,16)(13,24,37,22)(20,47,49,39)(21,43,51,58)(25,53,52,35)(26,40,57,59)(27,46,34,41)(30,36,50,62)(31,44,33,45)(48,63,55,67)(54,61,68,64)(56,66,60,65);(5,7)(6,13)(8,25)(9,30)(10,24)(11,22)(12,23)(14,39)(15,43)(16,38)(17,19)(18,37)(20,27)(21,31)(26,55)(28,52)(29,58)(32,50)(33,51)(34,49)(35,41)(36,44)(40,64)(42,47)(45,62)(46,53)(48,57)(54,56)(59,61)(60,68)(63,65)(66,67);(2,4)(5,8)(6,14)(7,20)(9,26)(10,27)(11,34)(12,28)(13,35)(15,40)(16,41)(17,46)(18,42)(19,39)(21,48)(22,25)(23,49)(24,52)(29,59)(30,54)(31,56)(32,57)(33,60)(36,61)(37,53)(38,47)(43,63)(44,65)(45,66)(50,68)(51,55)(58,67)(62,64);(5,9,10,31,12,32,11,33)(6,15,16,44,18,29,17,45)(7,21,22,30,23,51,24,50)(8,26,27,56,28,57,34,60)(13,36,19,43,37,62,38,58)(14,40,41,65,42,59,46,66)(20,48,25,54,49,55,52,68)(35,61,39,63,53,64,47,67)

Polycyclic presentation:

2^9|a^2=i,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325682);