Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2390 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)(14,20)(15,24)(17,18)(21,26)(22,23)(27,28);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,23,19,22,17,20)(15,25,24,27,26,16,21,28);(6,9)(7,11)(13,15,18,24,19,26,17,21)(14,16,23,28,22,25,20,27);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,23)(17,27)(18,28)(19,25)(20,26)(22,24);(6,9)(10,12)(14,22)(16,25)(20,23)(27,28)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6427743);