#148645

order	512
nilpotency class	4
isoclinism	isoclinic class of #136630
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1875
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11200
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

631 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)(3,14,18,9)(4,19)(5,25)(6,13)(7,32)(8,11)(10,22)(12,29)(15,45,46,61)(16,48,26,59)(17,36,31,35)(20,52,51,63)(21,53)(23,42)(24,57)(27,43)(28,40)(30,58)(33,47,44,64)(34,50,41,60)(37,49,62,55)(38,56)(39,54);(1,3)(2,9)(4,20)(5,26)(6,31)(7,17)(8,18)(10,37)(11,14)(12,41)(13,36)(15,23)(16,30)(19,52)(21,51)(22,55)(24,46)(25,48)(27,47)(28,50)(29,60)(32,35)(33,38)(34,40)(39,62)(42,45)(43,44)(49,54)(53,63)(56,64)(57,61)(58,59);(1,4,7,23,8,21,6,24)(2,10,32,38,11,39,13,43)(3,15,17,51,18,46,31,20)(5,27,29,22,30,56,28,54)(9,33,35,62,14,44,36,37)(12,42,58,53,40,57,25,19)(16,49,50,47,26,55,60,64)(34,52,48,45,41,63,59,61)

Polycyclic presentation:

2^9|a^2=g,c^2=dh,e^2=hi,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,h^a=hi,c^b=cf,e^b=ei,f^b=fi,g^b=gi,d^c=di,e^c=eh,f^c=fi,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, 55841);