#501993

order	512
nilpotency class	4
isoclinism	isoclinic class of #501992
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25012
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28245	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

513 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)(3,16)(4,23)(5,29)(6,15)(7,13)(8,11)(9,19)(10,27)(12,30)(14,32)(17,48)(18,44)(20,36)(21,38)(22,46)(24,57)(25,42)(26,41)(28,47)(31,43)(33,37)(34,45)(35,50)(39,53)(40,60)(49,55)(51,61)(52,64)(54,62)(56,59)(58,63);(1,3)(2,9,13,38)(4,24,27,58)(5,18)(6,19)(7,22)(8,33)(10,39)(11,36,14,44)(12,16,15,46)(17,25,51,59)(20,31)(21,30)(23,55)(26,49,28,53)(29,37,43,64)(32,52)(34,54,60,50)(35,56)(40,61)(41,63)(42,62)(45,48)(47,57);(1,4)(2,10)(3,17)(5,25)(6,26)(7,27)(8,34)(9,35)(11,40)(12,41)(13,23)(14,45)(15,47)(16,48)(18,49)(19,50)(20,53)(21,54)(22,51)(24,52)(28,30)(29,56)(31,59)(32,60)(33,58)(36,39)(37,63)(38,62)(42,43)(44,55)(46,61)(57,64);(1,5,30,32,7,31,6,8)(2,11,15,43,13,14,12,29)(3,18,21,52,22,20,19,33)(4,25,28,60,27,59,26,34)(9,36,46,64,38,44,16,37)(10,40,47,42,23,45,41,56)(17,49,54,24,51,53,50,58)(35,39,61,57,62,55,48,63)

Polycyclic presentation:

2^9|b^2=efh,d^2=ei,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,g^c=ghi,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, 409189);