#8926558

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

3714 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=h,e^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,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, 8833754);