#8926635

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,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, 8833831);