#479877

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	#17297
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

514 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)(3,13,17,42)(4,9,20,35)(5,21)(6,11)(7,12)(8,16,31,44)(10,25)(14,43)(15,48,50,36)(18,33,54,61)(19,34,26,39)(22,57,52,30)(23,37,55,56)(24,40)(27,38)(28,32)(29,51)(41,47)(45,63)(46,59)(49,58)(53,62)(60,64);(1,3)(2,8,12,31)(4,14,20,47)(5,22)(6,16)(7,17)(9,29)(10,36,21,48)(11,13,32,42)(15,27)(18,53,54,49)(19,46,26,51)(23,45,55,64)(24,50)(25,52)(28,44)(30,40,57,38)(33,60)(34,43)(35,59)(37,62)(39,41)(56,58)(61,63);(1,4,6,26,7,20,28,19)(2,9,32,34,12,35,11,39)(3,14,44,46,17,47,16,51)(5,23,27,54,25,55,24,18)(8,29,13,41,31,59,42,43)(10,37,38,33,21,56,40,61)(15,49,22,45,50,53,52,64)(30,60,36,62,57,63,48,58);(1,5,6,27,7,25,28,24)(2,10,32,38,12,21,11,40)(3,15,44,22,17,50,16,52)(4,18,26,23,20,54,19,55)(8,30,13,36,31,57,42,48)(9,33,34,56,35,61,39,37)(14,45,46,53,47,64,51,49)(29,58,41,60,59,62,43,63)

Polycyclic presentation:

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

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, 387073);