#149233

order	512
nilpotency class	4
isoclinism	isoclinic class of #13523
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2092
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11325
prime radical	C₂⁴	#135
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

471 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=eh,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,e^b=ei,f^b=fi,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, 56429);