#465307

order	512
nilpotency class	3
isoclinism	isoclinic class of #455198
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2620
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16518
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

751 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)(3,14)(4,9)(5,25)(6,13)(7,12)(8,17)(10,28)(11,20)(15,33)(16,46)(18,31)(19,51)(21,34)(22,40)(23,35)(24,36)(26,57)(27,38)(29,44)(30,59)(32,50)(37,54)(39,53)(41,45)(42,49)(43,60)(47,58)(48,56)(52,62)(55,61)(63,64);(1,3,7,18)(2,8)(4,19)(5,16,28,49)(6,17,20,50)(9,33,36,41)(10,30)(11,14)(12,32)(13,31)(15,22)(21,52)(23,45)(24,44)(25,56)(26,43)(27,48,53,59)(29,35,51,40)(34,60,61,63)(37,62,57,58)(38,42)(39,46)(47,55)(54,64);(1,4,20,22,7,24,6,23)(2,9,13,35,12,36,11,40)(3,15,17,19,18,45,50,44)(5,26,53,55,28,54,27,21)(8,29,14,41,32,51,31,33)(10,37,39,61,25,57,38,34)(16,47,48,43,49,52,59,64)(30,60,42,58,56,63,46,62);(1,5,6,27,7,28,20,53)(2,10,11,38,12,25,13,39)(3,16,17,48,18,49,50,59)(4,21,22,26,24,55,23,54)(8,30,14,42,32,56,31,46)(9,34,35,37,36,61,40,57)(15,43,44,47,45,64,19,52)(29,58,33,60,51,62,41,63)

Polycyclic presentation:

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

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, 372503);