#148063

order	512
nilpotency class	4
isoclinism	isoclinic class of #13334
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2011
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11049
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

699 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,e^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fi,h^a=hi,c^b=cf,f^b=fi,d^c=di,e^c=eh,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, 55259);