#385935

order	512
nilpotency class	3
isoclinism	isoclinic class of #385935
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

1359 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)(4,9,22,39)(5,23,26,41)(6,13)(7,12)(8,17)(10,25,42,29)(11,30)(15,44,46,35)(16,32,47,55)(18,33)(19,31,48,57)(20,51)(21,38,28,37)(24,43,49,40)(27,34)(36,52)(45,59)(50,62)(53,60)(54,61)(56,64)(58,63);(1,3)(2,8)(4,19)(5,24)(6,27)(7,18)(9,35)(10,40)(11,33)(12,34)(13,14)(15,21)(16,25)(17,30)(20,45)(22,48)(23,55)(26,49)(28,46)(29,47)(31,38)(32,41)(36,59)(37,57)(39,44)(42,43)(50,52)(51,58)(53,64)(54,63)(56,61)(60,62);(1,4,7,22)(2,9)(3,15,18,46)(5,20,26,53)(6,28,30,21)(8,31)(10,36)(11,38)(12,39)(13,37)(14,35)(16,45,47,64)(17,48,27,19)(23,51)(24,50,49,63)(25,54,29,52)(32,56)(33,44)(34,57)(40,62)(41,60)(42,61)(43,58)(55,59);(1,5)(2,10)(3,16)(4,20)(6,29)(7,26)(8,32)(9,36)(11,23)(12,42)(13,41)(14,43)(15,45)(17,49)(18,47)(19,50)(21,54)(22,53)(24,27)(25,30)(28,52)(31,56)(33,40)(34,55)(35,58)(37,60)(38,51)(39,61)(44,62)(46,64)(48,63)(57,59)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306448, #72306456, #72306467
maximal normal subgroup of	#72306448, #72306456, #72306467

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293131);