#386375

order	512
nilpotency class	3
isoclinism	isoclinic class of #386374
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#14427
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

983 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)(3,15,20,9)(4,21,24,41)(5,26)(6,13)(7,14)(8,12)(10,25,37,30)(11,27)(16,43)(17,46,49,57)(18,34,19,33)(22,31)(23,36,54,61)(28,39)(29,40)(32,47,38,48)(35,45)(42,50)(44,62)(51,55,60,56)(52,63)(53,59)(58,64);(1,3)(2,9)(4,22)(5,17)(6,19)(7,18)(8,20)(10,35)(11,38)(12,15)(13,34)(14,33)(16,25)(21,31)(23,53)(24,50)(26,57)(27,48)(28,47)(29,49)(30,45)(32,39)(36,59)(37,43)(40,46)(41,42)(44,55)(51,58)(52,54)(56,64)(60,62)(61,63);(1,4)(2,10,12,37)(3,16)(5,23)(6,25)(7,30)(8,24)(9,31,15,42)(11,36,39,61)(13,21,14,41)(17,44)(18,22)(19,50)(20,45)(26,51,40,60)(27,55)(28,56)(29,54)(32,58,38,62)(33,35,34,43)(46,63,57,59)(47,53)(48,52)(49,64);(1,5,8,29)(2,11,12,39)(3,17,20,49)(4,23,24,54)(6,28,7,27)(9,32,15,38)(10,36,37,61)(13,26,14,40)(16,44,45,64)(18,48,19,47)(21,51,41,60)(22,52,50,53)(25,56,30,55)(31,58,42,62)(33,57,34,46)(35,59,43,63)

Polycyclic presentation:

2^9|a^2=h,c^2=efg,d^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,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,f^d=fi,g^d=gi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293571);