#149066

order	512
nilpotency class	4
isoclinism	isoclinic class of #13391
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2033
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂	#11292
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

449 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,d^2=h,f^2=h,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=cf,d^b=dh,f^b=fh,g^b=gh,h^b=hi,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, 56262);