#8926661

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

3711 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=gh,e^2=g,f^2=h,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, 8833857);