#413936

order	512
nilpotency class	3
isoclinism	isoclinic class of #413935
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15159
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1349 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)(3,14)(4,9)(5,24)(6,12)(7,13)(8,17)(10,27)(11,29)(15,34)(16,45)(18,32)(19,49)(20,51)(21,37)(22,36)(23,38)(25,40)(26,39)(28,33)(30,43)(31,47)(35,52)(41,48)(42,57)(44,58)(46,55)(50,59)(53,56)(54,60)(61,63)(62,64);(1,3,7,18)(2,8,13,33)(4,19,23,43)(5,16,26,47)(6,28,29,17)(9,34,38,58)(10,31,40,45)(11,32,12,14)(15,21,44,22)(20,50,53,62)(24,55,39,41)(25,48,27,46)(30,37,49,36)(35,59,60,64)(42,54,61,52)(51,63,56,57);(1,4)(2,9)(3,15)(5,20)(6,21)(7,23)(8,30)(10,35)(11,36)(12,37)(13,38)(14,34)(16,42)(17,43)(18,44)(19,28)(22,29)(24,56)(25,52)(26,53)(27,54)(31,57)(32,58)(33,49)(39,51)(40,60)(41,59)(45,63)(46,50)(47,61)(48,62)(55,64);(1,5,7,26)(2,10)(3,16,18,47)(4,20,23,53)(6,27,29,25)(8,31)(9,35)(11,24)(12,39)(13,40)(14,41)(15,42,44,61)(17,48,28,46)(19,50,43,62)(21,54,22,52)(30,57)(32,55)(33,45)(34,59)(36,56)(37,51)(38,60)(49,63)(58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321132);