#427934

order	512
nilpotency class	3
isoclinism	isoclinic class of #427933
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15376
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1252 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)(3,8)(4,9)(5,20,22,41)(6,12)(7,11)(10,23,37,26)(13,32)(14,45,21,55)(15,30)(16,31)(17,49,51,54)(18,35)(19,34)(24,39)(25,40)(27,38)(28,43)(29,46,36,48)(33,52,60,53)(42,61,50,64)(44,58)(47,57)(56,63,59,62);(1,3)(2,8)(4,13)(5,21)(6,24)(7,15)(9,32)(10,36)(11,30)(12,39)(14,22)(16,27)(17,50)(18,47)(19,43)(20,45)(23,46)(25,44)(26,48)(28,34)(29,37)(31,38)(33,56)(35,57)(40,58)(41,55)(42,51)(49,61)(52,62)(53,63)(54,64)(59,60);(1,4,7,19)(2,9,11,34)(3,13,15,43)(5,17,22,51)(6,25,27,18)(8,28,30,32)(10,33,37,60)(12,40,38,35)(14,42,21,50)(16,47,24,44)(20,54,41,49)(23,53,26,52)(29,56,36,59)(31,58,39,57)(45,61,55,64)(46,63,48,62);(1,5)(2,10)(3,14,15,21)(4,17)(6,26)(7,22)(8,29,30,36)(9,33)(11,37)(12,41)(13,42,43,50)(16,48,24,46)(18,53)(19,51)(20,38)(23,27)(25,52)(28,56,32,59)(31,55,39,45)(34,60)(35,54)(40,49)(44,63,47,62)(57,61,58,64)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286186, #72286206, #72286235
maximal normal subgroup of	#72286186, #72286206, #72286235

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335130);