Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234485 |
| 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 | |
1091 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,39,44)(35,40,47,57)(36,48,37,42)(38,52,53,43)(41,51,59,50)(45,55,49,64)(46,61,63,56)(54,62,58,60);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(19,25)(20,27)(22,28)(23,30)(33,35)(34,40)(36,49)(37,45)(38,46)(39,47)(41,58)(42,55)(43,56)(44,57)(48,64)(50,62)(51,60)(52,61)(53,63)(54,59);(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32)(33,36,39,37)(34,41,44,59)(35,45,47,49)(38,51,53,50)(40,54,57,58)(42,52,48,43)(46,62,63,60)(55,56,64,61);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,25)(22,28)(26,31)(29,32)(33,37,39,36)(34,42,44,48)(35,45,47,49)(38,51,53,50)(40,55,57,64)(41,52,59,43)(46,60,63,62)(54,56,58,61);(2,8)(6,9)(7,16)(14,15)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(36,37)(41,59)(42,48)(45,49)(50,51)(54,58)(55,64)(60,62)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=fi,e^2=hi,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,e^c=ei,g^c=gi,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7207543);