#500895

order	512
nilpotency class	4
isoclinism	isoclinic class of #500894
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#24299
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3067	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

728 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=e,d^2=ei,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,g^b=gi,f^c=fhi,g^c=gi,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, 408091);