Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,11,12,10)(13,14,19,24)(15,20,26,16)(17,21,27,25)(18,23,28,22);(1,2,3,4)(6,9)(10,11)(13,15)(14,20)(16,24)(17,23)(18,25)(19,26)(21,28)(22,27);(5,7,8,12)(6,10,9,11)(13,16,18,17,19,20,28,27)(14,15,23,21,24,26,22,25);(13,17,18,20,19,27,28,16)(14,21,23,26,24,25,22,15);(13,18,19,28)(14,22,24,23)(15,25,26,21)(16,17,20,27)Polycyclic presentation:
2^9|a^2=gi,b^2=fg,c^2=gh,d^2=h,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,h^b=hi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6374008);