Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347544 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62831 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,7,12)(3,13,17,22)(4,9,18,24)(5,11,6,10)(8,15,23,16)(14,27,28,32)(19,26,20,25)(21,29,31,30)(33,34)(35,39)(36,40)(37,42)(38,41)(43,55)(44,51)(45,50)(46,53)(47,52)(48,54)(49,57)(56,63)(58,61)(59,62)(60,64);(1,3,7,17)(2,8,12,23)(4,14,18,28)(5,16,6,15)(9,21,24,31)(10,13,11,22)(19,30,20,29)(25,27,26,32)(33,35)(34,39)(36,43)(37,44)(38,45)(40,49)(41,50)(42,51)(46,56)(47,57)(48,58)(52,55)(53,60)(54,61)(59,64)(62,63);(1,4)(2,9)(3,14)(5,19)(6,20)(7,18)(8,21)(10,25)(11,26)(12,24)(13,27)(15,29)(16,30)(17,28)(22,32)(23,31)(33,36)(34,40)(35,43)(37,46)(38,47)(39,49)(41,52)(42,53)(44,56)(45,57)(48,59)(50,55)(51,60)(54,62)(58,64)(61,63);(4,18)(9,24)(14,28)(19,20)(21,31)(25,26)(27,32)(29,30)(34,41)(39,50)(40,52)(42,54)(49,55)(51,61)(53,62)(60,63);(1,5,7,6)(2,10,12,11)(3,15,17,16)(4,19,18,20)(8,22,23,13)(9,25,24,26)(14,29,28,30)(21,32,31,27)(33,37,38,48)(34,42,41,54)(35,44,45,58)(36,46,47,59)(39,51,50,61)(40,53,52,62)(43,56,57,64)(49,60,55,63)Polycyclic presentation:
2^9|a^2=h,b^2=h,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,d^c=dh,f^c=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254775);