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 | #25754 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2115 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,19)(15,23,24,20)(16,21,25,26)(17,22,27,28);(6,9)(10,12)(13,15,18,24)(14,20,19,23)(16,22,25,28)(17,26,27,21);(1,2,3,4)(13,16)(14,21)(15,17)(18,25)(19,26)(20,28)(22,23)(24,27);(13,17)(14,22)(15,16)(18,27)(19,28)(20,26)(21,23)(24,25);(5,7,8,11)(6,10,9,12)(15,24)(16,25)(20,23)(21,26)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,e^b=ei,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, 6352540);