#432666

order	512
nilpotency class	3
isoclinism	isoclinic class of #418871
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2697
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#15463
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1081 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,g^b=gi,h^b=hi,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, 339862);