#148301

order	512
nilpotency class	4
isoclinism	isoclinic class of #135850
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1854
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

645 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,d^2=h,f^2=i,g^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,g^b=gi,d^c=di,f^c=fi,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, 55497);