#437713

order	512
nilpotency class	3
isoclinism	isoclinic class of #437712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15610
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

873 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)(3,13)(4,9,19,32)(5,21,22,51)(6,12)(7,11)(8,16)(10,23,34,52)(14,30,40,27)(15,42,43,57)(17,36)(18,46)(20,33,25,37)(24,29)(26,35)(28,53,38,44)(31,54)(39,61)(41,58,45,56)(47,63)(48,60)(49,50)(55,62)(59,64);(1,3)(2,8,11,13)(4,14,19,40)(5,15,22,43)(6,24)(7,16)(9,30)(10,28)(12,36,35,29)(17,26)(18,47)(20,45,25,41)(21,42)(23,53,52,44)(27,32)(31,55,60,59)(33,56)(34,38)(37,58)(39,49)(46,61,50,63)(48,62)(51,57)(54,64);(1,4)(2,9,11,32)(3,14,16,40)(5,18)(6,25)(7,19)(8,27)(10,31,34,60)(12,37,35,33)(13,30)(15,39,43,47)(17,45,24,41)(20,26)(21,50,51,46)(22,49)(23,54)(28,55)(29,58)(36,56)(38,59)(42,63)(44,64,53,62)(48,52)(57,61);(1,5,7,22)(2,10)(3,15)(4,18,19,49)(6,23,26,52)(8,28,13,38)(9,31)(11,34)(12,21)(14,39)(16,43)(17,44)(20,48,25,54)(24,53)(27,55,30,59)(29,57,36,42)(32,60)(33,46)(35,51)(37,50)(40,47)(41,62)(45,64)(56,63,58,61)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 344909);