Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470867 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(15,22)(16,17)(19,27)(20,26)(23,28)(24,25);(2,4)(5,7)(6,9)(8,11)(10,12)(13,14)(15,23)(16,26)(17,20)(18,21)(19,24)(22,28)(25,27);(7,11)(9,12)(13,15,17,22,18,24,16,25)(14,19,20,27,21,23,26,28);(7,11)(9,12)(15,24)(19,23)(22,25)(27,28);(6,10)(9,12)(13,16,18,17)(14,20,21,26)(15,25,24,22)(19,27,23,28)Polycyclic presentation:
2^9|c^2=g,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6380199);