Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234437 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1109 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34)(35,44)(36,41)(37,42)(38,40)(39,46)(43,51)(45,56)(47,54)(48,53)(49,58)(50,55)(52,60)(57,62)(59,64)(61,63);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(19,25)(20,27)(22,28)(23,30)(33,35)(34,39)(36,48)(37,46)(38,47)(40,52)(41,56)(42,44)(43,54)(45,49)(50,61)(51,60)(53,58)(55,64)(57,63)(59,62);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(34,40)(36,49)(38,51)(39,52)(41,57)(42,43)(44,54)(45,48)(47,60)(53,64)(55,58)(56,63);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(19,25)(22,28)(26,31)(29,32)(33,36,37,49)(34,41,42,58)(35,45,46,48)(38,50,51,62)(39,53,44,56)(40,55,43,57)(47,59,60,61)(52,63,54,64);(2,10)(7,11)(8,9)(15,16)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(34,42)(35,46)(36,49)(40,43)(47,60)(50,62)(53,56)(63,64)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=fi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7207996);