#8417107

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3643 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)(3,10)(4,7)(5,12)(6,8)(9,14)(11,15)(13,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(34,39)(35,43)(36,46)(42,49)(45,57)(48,60)(53,55)(63,64);(1,3,7,6)(2,8,4,10)(5,11,14,13)(9,15,12,16)(19,25)(22,28)(26,31)(29,32)(34,39)(36,46)(40,51)(42,49)(44,47)(48,60)(54,59)(58,61);(1,2,7,4)(3,10,6,8)(5,9,14,12)(11,16,13,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,34)(35,40)(36,41)(37,49)(38,39)(42,50)(43,51)(44,53)(45,59)(46,52)(47,55)(48,62)(54,57)(56,60)(58,64)(61,63);(1,4,7,2)(3,8,6,10)(5,12,14,9)(11,15,13,16)(17,19,21,25)(18,22,24,28)(20,26,27,31)(23,29,30,32)(33,35)(34,40)(36,47)(37,45)(38,43)(39,51)(41,55)(42,54)(44,46)(48,61)(49,59)(50,57)(52,53)(56,64)(58,60)(62,63);(1,5)(2,9)(3,11)(4,12)(6,13)(7,14)(8,15)(10,16)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32)(33,36,38,46)(34,41,39,52)(35,44,43,47)(37,48,50,60)(40,53,51,55)(42,56,49,62)(45,58,57,61)(54,63,59,64);(1,6,7,3)(2,8,4,10)(5,13,14,11)(9,15,12,16)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,37,38,50)(34,42,39,49)(35,45,43,57)(36,48,46,60)(40,54,51,59)(41,56,52,62)(44,58,47,61)(53,63,55,64)

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=g,d^2=ghi,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dghi,e^b=egi,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, 8324303);