#419468

order	512
nilpotency class	3
isoclinism	isoclinic class of #419458
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2709
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15270
prime radical	#23414	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1316 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)(3,13,16,8)(4,9,20,34)(5,22,23,10)(6,25)(7,11)(12,28)(14,29)(15,36)(17,31,26,38)(18,46)(19,33)(21,39,27,35)(24,37,52,51)(30,44)(32,45)(40,42)(41,60,63,55)(43,56)(47,57,64,58)(48,62)(49,50)(53,61)(54,59);(1,3)(2,8)(4,18)(5,15)(6,26)(7,16)(9,32)(10,36)(11,13)(12,38)(14,21)(17,28)(19,47)(20,45)(22,30)(23,44)(24,53)(25,31)(27,42)(29,35)(33,58)(34,46)(37,56)(39,40)(41,48)(43,52)(49,64)(50,57)(51,61)(54,63)(55,62)(59,60);(1,4)(2,9,11,34)(3,14)(5,19)(6,27)(7,20)(8,29,13,40)(10,33,22,50)(12,39,25,35)(15,41)(16,42)(17,45)(18,26)(21,28)(23,49)(24,54)(30,55,36,60)(31,46,38,32)(37,62,51,59)(43,64)(44,63)(47,53)(48,52)(56,58,61,57);(1,5)(2,10)(3,15)(4,19)(6,24)(7,23)(8,30)(9,33)(11,22)(12,37)(13,36)(14,41)(16,44)(17,43)(18,47)(20,49)(21,48)(25,51)(26,53)(27,54)(28,52)(29,55)(31,56)(32,57)(34,50)(35,59)(38,61)(39,62)(40,60)(42,63)(45,64)(46,58)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326664);