Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437547 |
| 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 | C2×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2887 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)(14,20)(15,21)(22,27)(24,25);(2,4)(5,7,8,11)(6,9,10,12)(13,14,18,20)(15,23,21,28)(16,24,26,25)(17,22,19,27);(6,10)(9,12)(13,15,18,21)(14,16,20,26)(17,25,19,24)(22,28,27,23);(13,16)(14,21)(15,20)(17,23)(18,26)(19,28)(22,25)(24,27);(13,17,18,19)(14,22,20,27)(15,24,21,25)(16,23,26,28)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dh,h^b=hi,e^c=ei,h^c=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6347729);