#413889

order	512
nilpotency class	3
isoclinism	isoclinic class of #400688
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2544
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15159
prime radical	#23434	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1356 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)(5,26)(6,12)(7,13)(8,17)(9,24)(10,28)(11,30)(15,42)(16,46)(18,34)(19,33)(21,51)(22,53)(23,38)(25,37)(27,40)(29,39)(31,45)(32,49)(35,44)(36,56)(41,48)(43,60)(47,57)(50,54)(52,61)(55,58)(59,63)(62,64);(1,3)(2,8)(4,21)(5,16)(6,18)(7,19)(9,35)(10,32)(11,33)(12,14)(13,34)(15,24)(17,30)(20,42)(22,52)(23,45)(25,44)(26,57)(27,48)(28,47)(29,49)(31,37)(36,60)(38,51)(39,41)(40,46)(43,54)(50,61)(53,64)(55,62)(56,63)(58,59);(1,4)(2,9)(3,15)(5,22)(6,23)(7,25)(8,31)(10,36)(11,37)(12,20)(13,38)(14,35)(16,43)(17,44)(18,21)(19,45)(24,30)(26,58)(27,54)(28,55)(29,56)(32,59)(33,51)(34,42)(39,50)(40,53)(41,60)(46,61)(47,52)(48,63)(49,62)(57,64);(1,5,7,29)(2,10)(3,16,19,49)(4,22,25,56)(6,28,30,27)(8,32)(9,36)(11,26)(12,39)(13,40)(14,41)(15,43,45,62)(17,48,18,47)(20,50)(21,52,44,63)(23,55,24,54)(31,59)(33,57)(34,46)(35,60)(37,58)(38,53)(42,61)(51,64)

Polycyclic presentation:

2^9|d^2=gh,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,f^c=fi,g^c=gi,h^c=hi,e^d=ei,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, 321085);