#359668

order	512
nilpotency class	3
isoclinism	isoclinic class of #359665
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#13867
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62859	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3711 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251694, #72251709, #72251726, …
maximal normal subgroup of	#72251694, #72251709, #72251726, …
maximal quotient of	#72255925

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266864);