Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 | #35821 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2284 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)(13,14,17,20)(15,22,26,19)(16,21,27,18)(23,28,25,24);(5,7)(6,9)(8,12)(10,11)(13,15,17,26)(14,19,20,22)(16,24,27,28)(18,23,21,25);(1,2,3,4)(14,20)(15,23)(18,21)(19,28)(22,24)(25,26);(6,10)(9,11)(13,16)(14,21)(15,24)(17,27)(18,20)(19,25)(22,23)(26,28);(7,12)(9,11)(15,25)(16,27)(18,21)(19,28)(22,24)(23,26)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589924);