#9913335

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	#2699	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3766 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9820531);