#9912837

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3757 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=h,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^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, 9820033);