#148652

order	512
nilpotency class	4
isoclinism	isoclinic class of #135291
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1825
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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=dh,d^2=i,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,d^b=di,g^b=gi,d^c=di,e^c=ehi,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, 55848);