#148115

order	512
nilpotency class	4
isoclinism	isoclinic class of #13374
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2023
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

677 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=h,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=eh,f^b=fh,h^b=hi,d^c=di,e^c=ehi,f^c=fhi,g^c=gi,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, 55311);