Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6353061 |
| 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 | #62945 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2612 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)(3,12)(4,16)(5,10)(6,20)(7,11)(8,14)(9,18)(13,28)(15,23)(17,26)(19,25)(21,24)(22,29)(27,31)(30,32)(33,34,38,42)(35,39,45,51)(36,40,47,53)(37,41,48,54)(43,55,57,49)(44,50,58,61)(46,52,59,62)(56,63,64,60);(1,3,7,15)(2,8,11,24)(4,13,18,31)(5,14,6,21)(9,22,16,32)(10,23,20,12)(17,29,19,30)(25,27,26,28)(33,35)(34,39)(36,43)(37,44)(38,45)(40,49)(41,50)(42,51)(46,56)(47,57)(48,58)(52,60)(53,55)(54,61)(59,64)(62,63);(33,36,38,47)(34,40,42,53)(35,43,45,57)(37,46,48,59)(39,49,51,55)(41,52,54,62)(44,56,58,64)(50,60,61,63);(1,4)(2,9)(3,13)(5,19)(6,17)(7,18)(8,22)(10,26)(11,16)(12,27)(14,30)(15,31)(20,25)(21,29)(23,28)(24,32)(37,48)(41,54)(44,58)(46,59)(50,61)(52,62)(56,64)(60,63);(1,5,7,6)(2,10,11,20)(3,14,15,21)(4,17,18,19)(8,23,24,12)(9,25,16,26)(13,29,31,30)(22,27,32,28)(33,37,38,48)(34,41,42,54)(35,44,45,58)(36,46,47,59)(39,50,51,61)(40,52,53,62)(43,56,57,64)(49,60,55,63)Polycyclic presentation:
2^9|a^2=i,b^2=h,c^2=i,e^2=hi,f^2=h,b^a=bf,c^a=cg,d^a=dh,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ehi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6260422);