#8926656

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	C₂×#121	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:

(1,2,7,12)(3,11,6,8)(4,5,10,9)(13,16,15,14)(18,21)(23,26)(24,29)(30,32)(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);(1,3,7,6)(2,8,12,11)(4,15,10,13)(5,16,9,14)(20,28)(24,29)(25,31)(30,32)(34,40)(37,51)(41,53)(42,55)(44,56)(48,52)(57,64)(59,61);(1,4,7,10)(2,9,12,5)(3,13,6,15)(8,16,11,14)(17,18,22,21)(19,23,27,26)(20,24,28,29)(25,30,31,32)(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);(1,2)(3,8)(4,9)(5,10)(6,11)(7,12)(13,16)(14,15)(17,19,22,27)(18,23,21,26)(20,25,28,31)(24,30,29,32)(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);(1,5,7,9)(2,10,12,4)(3,14,6,16)(8,15,11,13)(17,20)(18,24)(19,25)(21,29)(22,28)(23,30)(26,32)(27,31)(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);(1,6,7,3)(2,11,12,8)(4,15,10,13)(5,16,9,14)(17,21,22,18)(19,26,27,23)(20,29,28,24)(25,32,31,30)(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=i,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, 8833852);