#8417288

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

3713 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)(7,11)(10,15)(14,16)(18,20)(23,25)(24,28)(30,31)(34,39)(35,43)(36,46)(42,49)(45,57)(48,60)(53,55)(63,64);(1,2,6,5)(3,7,9,11)(4,12,13,8)(10,16,15,14)(17,18,22,20)(19,25,27,23)(21,24,29,28)(26,31,32,30)(34,39)(36,46)(40,51)(42,49)(44,47)(48,60)(54,59)(58,61);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31)(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);(1,4,6,13)(2,8,5,12)(3,10,9,15)(7,14,11,16)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30)(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);(1,3)(2,7)(4,10)(5,11)(6,9)(8,14)(12,16)(13,15)(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);(1,5,6,2)(3,11,9,7)(4,12,13,8)(10,16,15,14)(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)(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|b^2=hi,c^2=g,d^2=hi,e^2=gi,f^2=gh,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, 8324484);