#8757387

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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=g,c^2=ghi,d^2=h,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eghi,f^b=fgi,d^c=dgi

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, 8664583);