#9481530

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	#392	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3637 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)(3,10)(4,7)(5,13)(6,9)(8,14)(11,12)(15,16)(17,18,23,19)(20,27,30,24)(21,25,22,26)(28,32,29,31)(33,34)(35,41)(36,42)(37,43)(38,44)(39,40)(45,47)(46,56)(48,57)(49,54)(50,58)(51,52)(53,55)(59,60)(61,63)(62,64);(1,3,6,7)(2,4,9,10)(5,11,14,15)(8,12,13,16)(18,19)(24,27)(25,26)(31,32)(34,40)(37,41)(42,54)(44,55)(47,50)(51,56)(57,64)(60,63);(1,2,6,9)(3,4,7,10)(5,13,14,8)(11,16,15,12)(17,19,23,18)(20,27,30,24)(21,26,22,25)(28,32,29,31)(33,35,39,43)(34,41,40,37)(36,45,49,58)(38,52,53,46)(42,47,54,50)(44,51,55,56)(48,61,62,59)(57,63,64,60);(1,2)(3,4)(5,8)(6,9)(7,10)(11,12)(13,14)(15,16)(17,20,23,30)(18,24,19,27)(21,28,22,29)(25,31,26,32)(33,36)(34,42)(35,45)(37,50)(38,48)(39,49)(40,54)(41,47)(43,58)(44,57)(46,59)(51,63)(52,61)(53,62)(55,64)(56,60);(1,4)(2,7)(3,9)(5,12)(6,10)(8,15)(11,13)(14,16)(17,21,23,22)(18,25,19,26)(20,28,30,29)(24,31,27,32)(33,37,39,41)(34,43,40,35)(36,47,49,50)(38,51,53,56)(42,45,54,58)(44,52,55,46)(48,60,62,63)(57,59,64,61);(1,5,6,14)(2,8,9,13)(3,11,7,15)(4,12,10,16)(17,22,23,21)(18,26,19,25)(20,29,30,28)(24,32,27,31)(33,38,39,53)(34,44,40,55)(35,46,43,52)(36,48,49,62)(37,51,41,56)(42,57,54,64)(45,59,58,61)(47,60,50,63)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388726);