#148636

order	512
nilpotency class	4
isoclinism	isoclinic class of #13400
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2034
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

625 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=dh,e^2=h,f^2=i,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,f^c=fi,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, 55832);