Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433049 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2615 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1803 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,16)(14,17)(15,18)(19,20)(21,22,26,28)(23,30,32,36)(24,29,25,35)(27,33,31,34);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20)(21,23,25,34,26,32,24,33)(22,27,29,30,28,31,35,36);(2,4)(7,13)(10,16)(14,19)(17,20)(21,22)(23,31)(24,35)(25,29)(26,28)(27,32)(30,33)(34,36);(6,11)(10,16)(12,18)(17,20)(22,28)(23,32)(29,35)(33,34);(5,8,9,15)(6,12,11,18)(7,14,13,19)(10,17,16,20)(21,24,26,25)(22,29,28,35)(23,33,32,34)(27,30,31,36)Polycyclic presentation:
2^9|a^2=i,b^2=f,e^2=h,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,d^b=di,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343978);