Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667556 |
| 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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2392 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)(7,11)(8,10)(9,12)(13,14,19,20)(15,23,26,21)(16,18,22,17)(24,25,27,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,21)(16,25)(17,27)(18,24)(19,26)(20,23)(22,28);(1,2,3,4)(13,16)(14,18)(15,24)(17,20)(19,22)(21,28)(23,25)(26,27);(6,10)(9,11)(13,17)(14,22)(15,24)(16,20)(18,19)(21,25)(23,28)(26,27);(7,12)(9,11)(13,18,19,17)(14,22,20,16)(15,25,26,28)(21,24,23,27)Polycyclic presentation:
2^9|a^2=i,c^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589447);