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 | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2693 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(14,20)(15,22)(17,19)(21,23)(24,26)(27,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,14,18,20)(15,21,24,27)(16,17,25,19)(22,28,26,23);(2,4)(13,15,18,24)(14,21,20,27)(16,23,25,28)(17,22,19,26);(2,4)(13,16)(14,17)(15,23)(18,25)(19,20)(21,22)(24,28)(26,27);(13,17,18,19)(14,16,20,25)(15,22,24,26)(21,23,27,28)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=dh,e^b=ei,g^c=gi,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345346);