Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7234321 |
| 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 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1089 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 64 points:
(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18,21,24)(19,22,25,28)(20,23,27,30)(26,29,31,32)(33,34,39,43)(35,44,48,59)(36,41,50,57)(37,42,38,52)(40,46,54,47)(45,58,61,63)(49,55,51,56)(53,62,64,60);(1,3,5,10)(2,6,8,9)(4,11,13,16)(7,14,12,15)(17,19)(18,22)(20,26)(21,25)(23,29)(24,28)(27,31)(30,32);(19,25)(20,27)(22,28)(23,30)(33,35,37,46,39,48,38,47)(34,40,52,59,43,54,42,44)(36,45,51,62,50,61,49,60)(41,53,55,63,57,64,56,58);(3,10)(6,9)(11,16)(14,15)(17,20,21,27)(18,23,24,30)(19,26,25,31)(22,29,28,32)(33,36)(34,41)(35,45)(37,51)(38,49)(39,50)(40,53)(42,56)(43,57)(44,58)(46,62)(47,60)(48,61)(52,55)(54,64)(59,63);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(19,25)(22,28)(26,31)(29,32)(33,37,39,38)(34,42,43,52)(35,46,48,47)(36,49,50,51)(40,44,54,59)(41,55,57,56)(45,60,61,62)(53,63,64,58)Polycyclic presentation:
2^9|a^2=fhi,b^2=f,c^2=gi,d^2=h,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7200357);