#148329

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	#11096
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:

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

Polycyclic presentation:

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

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, 55525);