#148137

order	512
nilpotency class	4
isoclinism	isoclinic class of #13323
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1999
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11066
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=i,e^2=hi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,e^c=eh,f^c=fhi,h^c=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55333);