#8755422

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

3637 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=ghi,d^2=ghi,e^2=gh,f^2=g,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, 8662618);