#8926609

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

3703 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)(17,18,23,27)(19,21,30,24)(20,31,29,25)(22,32,28,26)(33,34,39,37)(35,40,46,44)(36,41,48,49)(38,42,52,50)(43,53,58,59)(45,54,51,56)(47,55,61,62)(57,63,60,64);(5,13)(8,12)(10,16)(14,15)(17,19,23,30)(18,24,27,21)(20,22,29,28)(25,32,31,26)(34,37)(40,44)(41,49)(42,50)(53,59)(54,56)(55,62)(63,64);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,20,23,29)(18,25,27,31)(19,28,30,22)(21,32,24,26)(33,35,39,46)(34,40,37,44)(36,43,48,58)(38,51,52,45)(41,53,49,59)(42,56,50,54)(47,60,61,57)(55,64,62,63);(1,4,6,2)(3,9,11,7)(5,12,13,8)(10,15,16,14)(17,21)(18,19)(20,32)(22,31)(23,24)(25,28)(26,29)(27,30)(33,36,39,48)(34,41,37,49)(35,43,46,58)(38,47,52,61)(40,53,44,59)(42,55,50,62)(45,57,51,60)(54,63,56,64);(1,5,6,13)(2,8,4,12)(3,10,11,16)(7,14,9,15)(17,19)(18,21)(20,28)(22,29)(23,30)(24,27)(25,32)(26,31)(33,37,39,34)(35,44,46,40)(36,41,48,49)(38,50,52,42)(43,53,58,59)(45,56,51,54)(47,55,61,62)(57,63,60,64);(1,3)(2,7)(4,9)(5,10)(6,11)(8,14)(12,15)(13,16)(17,22,23,28)(18,26,27,32)(19,29,30,20)(21,31,24,25)(33,38,39,52)(34,42,37,50)(35,45,46,51)(36,47,48,61)(40,54,44,56)(41,55,49,62)(43,57,58,60)(53,63,59,64)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=g,d^2=gh,e^2=gh,f^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, 8833805);