#417687

order	512
nilpotency class	3
isoclinism	isoclinic class of #417686
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15257
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2318 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)(4,10)(5,23)(6,14)(7,12)(8,13)(9,18)(11,27)(16,44)(17,31)(19,40)(20,51)(21,41)(22,36)(24,43)(25,39)(26,38)(28,32)(29,35)(30,48)(33,46)(34,53)(37,50)(42,47)(45,58)(49,56)(52,60)(54,57)(55,61)(59,64)(62,63);(1,3)(2,9)(4,16)(5,24)(6,28)(7,18)(8,19)(10,33)(11,37)(12,40)(13,32)(14,15)(17,25)(20,52)(21,48)(22,47)(23,56)(26,50)(27,49)(29,46)(30,36)(31,38)(34,60)(35,42)(39,43)(41,44)(45,53)(51,59)(54,63)(55,64)(57,58)(61,62);(1,4)(2,10)(3,16)(5,20)(6,21)(7,29)(8,22)(9,30)(11,34)(12,35)(13,36)(14,41)(15,42)(17,45)(18,46)(19,47)(23,57)(24,52)(25,53)(26,54)(27,55)(28,48)(31,58)(32,33)(37,62)(38,51)(39,61)(40,44)(43,60)(49,64)(50,63)(56,59);(1,5)(2,11)(3,17)(4,20)(6,27)(7,25)(8,26)(9,31)(10,34)(12,23)(13,39)(14,38)(15,43)(16,45)(18,50)(19,49)(21,55)(22,54)(24,28)(29,53)(30,58)(32,56)(33,59)(35,57)(36,61)(37,40)(41,51)(42,60)(44,62)(46,63)(47,64)(48,52)

Polycyclic presentation:

2^9|e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^c=hi,e^d=ei,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, 324883);