#462893

order	512
nilpotency class	3
isoclinism	isoclinic class of #460970
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2759
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16454
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

662 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)(6,12)(7,13)(8,14)(9,18,34,19)(10,25)(11,30)(16,32,47,53)(17,48,52,33)(22,43,46,35)(23,36)(24,38)(26,37)(28,39)(29,41)(31,40)(44,50,62,51)(45,63,64,58)(49,57,54,61)(55,59)(56,60);(1,3)(2,9,14,34)(4,22)(5,17)(6,18)(7,19)(8,20)(10,35,38,43)(11,33,41,48)(12,15,13,42)(16,24)(21,53,37,32)(23,54)(25,47)(26,46)(27,44,40,62)(28,45)(29,50)(30,51)(31,52)(36,58,60,63)(39,61,59,57)(49,56)(55,64);(1,4)(2,10)(3,16)(5,28)(6,25)(7,24)(8,26)(9,32)(11,39)(12,37)(13,21)(14,38)(15,43)(17,49)(18,22)(19,46)(20,47)(23,29)(27,36)(30,56)(31,55)(33,58)(34,53)(35,42)(40,60)(41,59)(44,57)(45,50)(48,63)(51,64)(52,54)(61,62);(1,5,7,30,8,31,6,29)(2,11,12,27,14,41,13,40)(3,17,19,51,20,52,18,50)(4,23,25,55,26,56,24,28)(9,33,15,44,34,48,42,62)(10,36,21,39,38,60,37,59)(16,45,22,54,47,64,46,49)(32,57,35,58,53,61,43,63)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 370089);