#394357

order	512
nilpotency class	3
isoclinism	isoclinic class of #393928
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14716
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2544 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)(3,9)(4,10)(5,23)(6,12)(7,27)(8,13)(11,24)(14,40)(15,43)(16,31)(17,32)(18,47)(19,49)(20,51)(21,35)(22,36)(25,38)(26,37)(28,39)(29,42)(30,45)(33,46)(34,54)(41,62)(44,57)(48,56)(50,55)(52,60)(53,59)(58,63)(61,64);(1,3)(2,9)(4,19,21,46)(5,15)(6,17)(7,18)(8,16)(10,33,35,49)(11,30)(12,32)(13,31)(14,22,42,26)(20,50,52,64)(23,43)(24,45)(25,44)(27,47)(28,48)(29,36,40,37)(34,58,59,62)(38,57)(39,56)(41,53,63,54)(51,61,60,55);(1,4)(2,10)(3,14)(5,20)(6,26)(7,22)(8,21)(9,29)(11,34)(12,37)(13,35)(15,41)(16,42)(17,46)(18,19)(23,51)(24,54)(25,52)(27,36)(28,53)(30,55)(31,40)(32,49)(33,47)(38,60)(39,59)(43,58)(44,63)(45,64)(48,50)(56,61)(57,62);(1,5)(2,11)(3,15)(4,20)(6,24)(7,28)(8,25)(9,30)(10,34)(12,38)(13,39)(14,41)(16,44)(17,45)(18,48)(19,50)(21,52)(22,53)(23,27)(26,54)(29,55)(31,56)(32,57)(33,58)(35,59)(36,51)(37,60)(40,61)(42,63)(43,47)(46,64)(49,62)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301553);