#148289

order	512
nilpotency class	4
isoclinism	isoclinic class of #134702
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1809
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11095
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

651 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)(3,15,19,42)(4,20)(5,24)(6,14)(7,12)(8,13)(9,17,32,49)(10,22)(11,26)(16,43,46,63)(18,50,28,41)(21,53)(23,39)(25,37)(27,36)(29,35)(30,58,33,60)(31,56,38,48)(34,55)(40,47,61,45)(44,59,64,51)(52,54)(57,62);(1,3,5,17,7,18,26,48,8,19,27,49,6,28,25,56)(2,9,11,15,14,31,24,41,13,32,37,42,12,38,36,50)(4,16,21,44,22,47,54,60,23,46,55,64,29,45,57,58)(10,33,34,61,20,51,52,43,35,30,53,40,39,59,62,63);(1,4)(2,10)(3,16)(5,21)(6,29)(7,22)(8,23)(9,30)(11,34)(12,39)(13,35)(14,20)(15,40)(17,44)(18,47)(19,46)(24,52)(25,57)(26,54)(27,55)(28,45)(31,59)(32,33)(36,62)(37,53)(38,51)(41,63)(42,61)(43,50)(48,60)(49,64)(56,58)

Polycyclic presentation:

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

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, 55485);