#149059

order	512
nilpotency class	4
isoclinism	isoclinic class of #13545
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2101
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:

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

Polycyclic presentation:

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

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, 56255);