#385913

order	512
nilpotency class	3
isoclinism	isoclinic class of #385912
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14417
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1365 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)(4,19,24,37)(5,10,27,42)(6,12)(7,13)(8,17)(9,23,38,22)(11,30)(15,31,46,49)(16,43,47,39)(18,33)(20,35,45,58)(21,52)(25,56,48,32)(26,41,29,40)(28,34)(36,53)(44,60)(50,55)(51,62)(54,61)(57,64)(59,63);(1,3,7,18)(2,8,13,34)(4,20,24,45)(5,25,27,48)(6,28,30,17)(9,35,38,58)(10,39,42,43)(11,33,12,14)(15,23,46,22)(16,29,47,26)(19,49,37,31)(21,44,55,64)(32,41,56,40)(36,60,61,57)(50,62,52,59)(51,54,63,53);(1,4,7,24)(2,9)(3,15,18,46)(5,21,27,55)(6,23,30,22)(8,31)(10,36)(11,37)(12,19)(13,38)(14,35)(16,44,47,64)(17,45,28,20)(25,51,48,63)(26,54,29,53)(32,57)(33,58)(34,49)(39,62)(40,50)(41,52)(42,61)(43,59)(56,60);(1,5)(2,10)(3,16)(4,21)(6,29)(7,27)(8,32)(9,36)(11,41)(12,40)(13,42)(14,43)(15,44)(17,48)(18,47)(19,50)(20,51)(22,54)(23,53)(24,55)(25,28)(26,30)(31,57)(33,39)(34,56)(35,59)(37,52)(38,61)(45,63)(46,64)(49,60)(58,62)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306392, #72306448, #72306453
maximal normal subgroup of	#72306392, #72306448, #72306453

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293109);