Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2175 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)(7,11)(8,10)(9,12)(15,24)(17,27)(19,28)(21,25);(5,7)(6,9)(8,12)(10,11)(13,14)(15,25)(16,20)(17,19)(18,22)(21,24)(23,26)(27,28);(2,4)(7,12)(9,11)(13,15,26,27,18,24,16,17)(14,19,20,21,22,28,23,25);(6,10)(9,11)(13,16,18,26)(14,20,22,23)(15,27,24,17)(19,25,28,21);(13,17,16,24,18,27,26,15)(14,21,23,19,22,25,20,28)Polycyclic presentation:
2^9|c^2=fhi,d^2=i,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717345);