#474260

order	512
nilpotency class	3
isoclinism	isoclinic class of #455699
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2797
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#16929
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

619 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)(4,21)(5,27,30,11)(6,13)(7,12)(8,14)(9,18)(10,24)(16,44)(17,48,52,43)(19,35)(20,34)(22,36)(23,40,54,57)(25,39)(26,38)(28,37,55,63)(29,41,31,42)(32,47)(33,50,56,51)(45,64,58,59)(46,60)(49,61,53,62);(1,3,7,18,8,20,6,19)(2,9,12,34,14,35,13,15)(4,22,24,47,26,46,25,16)(5,17,31,50,30,52,29,51)(10,36,38,32,39,60,21,44)(11,33,41,48,27,56,42,43)(23,53,28,58,54,49,55,45)(37,62,57,59,63,61,40,64);(1,4)(2,10)(3,16)(5,28)(6,25)(7,24)(8,26)(9,32)(11,40)(12,38)(13,21)(14,39)(15,36)(17,49)(18,22)(19,46)(20,47)(23,29)(27,57)(30,55)(31,54)(33,61)(34,60)(35,44)(37,41)(42,63)(43,59)(45,50)(48,64)(51,58)(52,53)(56,62);(1,5,6,29,8,30,7,31)(2,11,12,41,14,27,13,42)(3,17,19,51,20,52,18,50)(4,23,24,28,26,54,25,55)(9,33,34,48,35,56,15,43)(10,37,21,40,39,63,38,57)(16,45,22,53,47,58,46,49)(32,59,36,62,44,64,60,61)

Polycyclic presentation:

2^9|a^2=eg,b^2=efgh,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,g^d=gi,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, 381456);