#148703

order	512
nilpotency class	4
isoclinism	isoclinic class of #137786
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1899
commutator	C₂³×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂²×#41	C₂³
socle	C₂	#11203
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×#568 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

635 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)(4,19,23,50)(5,24)(6,26,28,12)(7,11)(8,13)(9,16)(10,22)(15,41,44,63)(17,46,47,40)(18,33)(20,51,42,60)(21,49,30,48)(25,39,27,55)(29,34)(31,59)(32,45,56,57)(35,43)(36,58)(37,53)(38,54)(52,61)(62,64);(1,3,5,16,8,18,7,29)(2,9,11,14,13,34,24,33)(4,20,21,15,23,42,30,44)(6,17,25,45,28,47,27,57)(10,35,36,61,38,62,37,31)(12,32,39,40,26,56,55,46)(19,41,48,51,50,63,49,60)(22,52,53,43,54,59,58,64);(1,4,5,21,8,23,7,30)(2,10,11,36,13,38,24,37)(3,15,16,42,18,44,29,20)(6,22,25,53,28,54,27,58)(9,31,14,35,34,61,33,62)(12,19,39,48,26,50,55,49)(17,43,45,59,47,64,57,52)(32,60,40,41,56,51,46,63)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55899);