#361317

order	512
nilpotency class	3
isoclinism	isoclinic class of #361316
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13894
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1333 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)(3,8,15,29)(4,17,21,9)(5,10,22,34)(6,12)(7,11)(13,39)(14,42)(16,30,24,37)(18,46)(19,47)(20,38,25,33)(23,35,51,58)(26,36)(27,41)(28,44)(31,45)(32,49)(40,54,62,61)(43,59)(48,56,63,64)(50,60)(52,55)(53,57);(1,3)(2,8,11,29)(4,18)(5,14)(6,24)(7,15)(9,31,17,46)(10,28,34,42)(12,37,36,30)(13,20)(16,26)(19,48)(21,45)(22,44)(23,52)(25,41)(27,38,39,33)(32,56,47,64)(35,59,58,55)(40,50)(43,51)(49,63)(53,62)(54,60,61,57);(1,4)(2,9)(3,13)(5,19)(6,25)(7,21)(8,27)(10,32)(11,17)(12,38)(14,40)(15,41)(16,45)(18,24)(20,26)(22,49)(23,53)(28,54)(29,39)(30,46)(31,37)(33,36)(34,47)(35,60)(42,61)(43,63)(44,62)(48,52)(50,51)(55,64)(56,59)(57,58);(1,5,7,22)(2,10)(3,14,15,44)(4,19,21,49)(6,23,26,51)(8,28)(9,32)(11,34)(12,35)(13,40,41,62)(16,43,24,52)(17,47)(18,48,45,63)(20,50,25,53)(27,54)(29,42)(30,55)(31,56)(33,57)(36,58)(37,59)(38,60)(39,61)(46,64)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268513);