#8417046

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=gh,d^2=i,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8324242);