#148615

order	512
nilpotency class	4
isoclinism	isoclinic class of #13389
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11199
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

623 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)(3,14,18,35)(4,19)(5,11)(6,26)(7,30)(8,13)(9,16,34,24)(10,22)(12,28)(15,44,20,54)(17,33,50,63)(21,52)(23,53)(25,41)(27,37)(29,59)(31,40)(32,60,36,49)(38,56)(39,61)(42,55,64,46)(43,57,58,48)(45,51,47,62);(1,3)(2,9)(4,20)(5,24)(6,17)(7,16)(8,18)(10,36)(11,35)(12,33)(13,34)(14,30)(15,23)(19,51)(21,45)(22,55)(25,57)(26,58)(27,60)(28,48)(29,50)(31,47)(32,38)(37,42)(39,64)(40,54)(41,63)(43,59)(44,52)(46,56)(49,61)(53,62);(1,4,5,21,8,23,7,31)(2,10,11,37,13,38,30,39)(3,15,16,45,18,20,24,47)(6,27,25,56,29,61,28,22)(9,32,14,42,34,36,35,64)(12,40,26,19,41,52,59,53)(17,49,48,46,50,60,57,55)(33,44,43,51,63,54,58,62)

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=di,g^b=gi,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, 55811);