#461122

order	512
nilpotency class	3
isoclinism	isoclinic class of #461121
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16380
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

849 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)(3,15,20,42)(4,21)(5,27,31,40)(6,12)(7,13)(8,14)(9,18,34,19)(10,25)(11,30,41,29)(16,32,47,52)(17,33)(22,43,46,35)(23,54,56,36)(24,38)(26,37)(28,53,55,39)(44,50)(45,59)(48,63)(49,60)(51,61)(57,62)(58,64);(1,3)(2,9,14,34)(4,22)(5,17)(6,18)(7,19)(8,20)(10,35,38,43)(11,33,41,61)(12,15,13,42)(16,24)(21,52,37,32)(23,48)(25,47)(26,46)(27,44,40,60)(28,57)(29,49)(30,50)(31,51)(36,62,54,59)(39,58,53,63)(45,55)(56,64);(1,4,8,26)(2,10,14,38)(3,16,20,47)(5,28,31,55)(6,25,7,24)(9,32,34,52)(11,39,41,53)(12,37,13,21)(15,43,42,35)(17,48,51,64)(18,22,19,46)(23,30,56,29)(27,54,40,36)(33,59,61,62)(44,63,60,58)(45,50,57,49);(1,5)(2,11)(3,17)(4,23)(6,30)(7,29)(8,31)(9,33)(10,36)(12,27)(13,40)(14,41)(15,44)(16,45)(18,50)(19,49)(20,51)(21,53)(22,48)(24,55)(25,28)(26,56)(32,58)(34,61)(35,62)(37,39)(38,54)(42,60)(43,59)(46,64)(47,57)(52,63)

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368318);