#9481469

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	#57248	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3767 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)(3,7)(4,11)(5,6)(8,12)(9,13)(10,16)(14,15)(17,18,23,21)(19,27,29,24)(20,30,31,25)(22,32,28,26)(33,34,39,37)(35,43,44,40)(36,47,50,41)(38,51,53,42)(45,54,48,56)(46,55,52,58)(49,57,62,60)(59,64,61,63);(3,9)(7,13)(10,15)(14,16)(17,19,23,29)(18,24,21,27)(20,28,31,22)(25,26,30,32)(35,44)(40,43)(45,48)(46,52)(54,56)(55,58)(59,61)(63,64);(2,6)(7,13)(8,11)(14,16)(17,20,23,31)(18,25,21,30)(19,22,29,28)(24,32,27,26)(33,35)(34,40)(36,48)(37,43)(38,52)(39,44)(41,56)(42,58)(45,50)(46,53)(47,54)(49,59)(51,55)(57,63)(60,64)(61,62);(1,2)(3,7)(4,8)(5,6)(9,13)(10,14)(11,12)(15,16)(17,20)(18,25)(19,28)(21,30)(22,29)(23,31)(24,26)(27,32)(33,36,39,50)(34,41,37,47)(35,45,44,48)(38,49,53,62)(40,54,43,56)(42,57,51,60)(46,59,52,61)(55,63,58,64);(1,3,5,9)(2,7,6,13)(4,10,12,15)(8,14,11,16)(17,21,23,18)(19,24,29,27)(20,30,31,25)(22,32,28,26)(33,37,39,34)(35,43,44,40)(36,41,50,47)(38,51,53,42)(45,54,48,56)(46,58,52,55)(49,57,62,60)(59,63,61,64);(1,4)(2,8)(3,10)(5,12)(6,11)(7,14)(9,15)(13,16)(17,22,23,28)(18,26,21,32)(19,20,29,31)(24,30,27,25)(33,38)(34,42)(35,46)(36,49)(37,51)(39,53)(40,55)(41,57)(43,58)(44,52)(45,59)(47,60)(48,61)(50,62)(54,63)(56,64)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=g,d^2=gi,e^2=h,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, 9388665);