#405286

order	512
nilpotency class	3
isoclinism	isoclinic class of #395586
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2411
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14911
prime radical	#60398	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2184 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)(3,13)(4,9)(5,10)(6,24)(7,12)(8,16)(11,27)(14,38)(15,39)(17,36)(18,45)(19,47)(20,37)(21,33)(22,50)(23,35)(25,30)(26,32)(28,44)(29,42)(31,49)(34,53)(40,61)(41,55)(43,58)(46,60)(48,57)(51,56)(52,59)(54,62)(63,64);(1,3)(2,8)(4,18,21,14)(5,15)(6,25)(7,17)(9,28,33,55)(10,29)(11,36)(12,30)(13,24)(16,27)(19,46,49,40)(20,41,26,44)(22,51)(23,43)(31,54,47,64)(32,38,37,45)(34,58)(35,56)(39,50)(42,53)(48,62,52,63)(57,60,59,61);(1,4)(2,9)(3,14)(5,19)(6,26)(7,21)(8,28)(10,31)(11,37)(12,33)(13,38)(15,40)(16,44)(17,18)(20,27)(22,52)(23,49)(24,32)(25,41)(29,54)(30,55)(34,59)(35,47)(36,45)(39,60)(42,63)(43,46)(48,53)(50,57)(51,62)(56,64)(58,61);(1,5)(2,10)(3,15)(4,19)(6,22)(7,23)(8,29)(9,31)(11,34)(12,35)(13,39)(14,40)(16,42)(17,43)(18,46)(20,48)(21,49)(24,50)(25,51)(26,52)(27,53)(28,54)(30,56)(32,57)(33,47)(36,58)(37,59)(38,60)(41,62)(44,63)(45,61)(55,64)

Polycyclic presentation:

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

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, 312482);