#417703

order	512
nilpotency class	3
isoclinism	isoclinic class of #417702
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15257
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2295 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324899);