#471045

order	512
nilpotency class	3
isoclinism	isoclinic class of #459299
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2752
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16738
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

642 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)(3,13,17,8)(4,18)(5,10)(6,25)(7,11)(9,21)(12,26)(14,38,19,46)(15,36,42,28)(16,29,43,37)(20,49)(22,45)(23,53)(24,34)(27,40,30,48)(31,58)(32,52)(33,50)(35,54)(39,61,47,60)(41,57,63,59)(44,51)(55,64,56,62);(1,3)(2,8)(4,19,22,14)(5,15)(6,16)(7,17)(9,30,32,27)(10,28)(11,13)(12,29)(18,38,45,46)(20,47,51,39)(21,48,52,40)(23,41)(24,42)(25,37)(26,43)(31,56,33,55)(34,36)(35,57)(44,60,49,61)(50,64,58,62)(53,59)(54,63);(1,4)(2,9,11,32)(3,14)(5,20)(6,21)(7,22)(8,27,13,30)(10,33,34,31)(12,18,25,45)(15,39)(16,40)(17,19)(23,50)(24,51)(26,52)(28,56,36,55)(29,46,37,38)(35,49,53,44)(41,62)(42,47)(43,48)(54,58)(57,60,59,61)(63,64);(1,5,6,23,7,24,26,54)(2,10,25,53,11,34,12,35)(3,15,16,41,17,42,43,63)(4,20,21,50,22,51,52,58)(8,28,37,59,13,36,29,57)(9,31,45,49,32,33,18,44)(14,39,40,62,19,47,48,64)(27,55,38,60,30,56,46,61)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=e,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,d^c=de,h^c=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, 378241);