#405287

order	512
nilpotency class	3
isoclinism	isoclinic class of #394043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2613
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14911
prime radical	#60387	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2158 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)(4,20)(5,11)(6,29)(7,12)(8,13)(9,17)(10,24)(15,42)(16,41)(18,33)(19,32)(21,50)(22,52)(23,37)(25,36)(26,39)(27,38)(28,40)(30,45)(31,46)(34,44)(35,53)(43,56)(47,57)(48,58)(49,54)(51,59)(55,60)(61,63)(62,64);(1,3)(2,9)(4,21,25,44)(5,16)(6,18)(7,17)(8,19)(10,34,37,50)(11,31)(12,32)(13,33)(14,29)(15,23,45,24)(20,42,36,30)(22,51,54,62)(26,48)(27,46)(28,47)(35,59,60,64)(38,57)(39,41)(40,58)(43,53,63,55)(49,61,52,56);(1,4)(2,10)(3,15)(5,22)(6,23)(7,24)(8,25)(9,30)(11,35)(12,36)(13,37)(14,34)(16,43)(17,44)(18,21)(19,45)(20,29)(26,53)(27,55)(28,54)(31,56)(32,50)(33,42)(38,52)(39,49)(40,60)(41,59)(46,62)(47,63)(48,51)(57,64)(58,61);(1,5)(2,11)(3,16)(4,22)(6,26)(7,27)(8,28)(9,31)(10,35)(12,38)(13,40)(14,41)(15,43)(17,46)(18,48)(19,47)(20,49)(21,51)(23,53)(24,55)(25,54)(29,39)(30,56)(32,57)(33,58)(34,59)(36,52)(37,60)(42,61)(44,62)(45,63)(50,64)

Polycyclic presentation:

2^9|b^2=h,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^c=fi,g^c=gi,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, 312483);