#479787

order	512
nilpotency class	3
isoclinism	isoclinic class of #477835
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2793
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#17247
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

494 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386983);