Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| 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 ◁ #2397 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2427 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)(6,9)(7,11)(13,14,18,19)(15,22,25,20)(16,26,17,21)(23,28,27,24);(5,6)(7,10)(8,9)(11,12)(13,15,18,25)(14,20,19,22)(16,23,17,27)(21,28,26,24);(1,2,3,4)(14,19)(15,22)(20,25)(21,26)(23,24)(27,28);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,23)(17,18)(19,26)(20,28)(22,24)(25,27);(6,9)(10,12)(13,17,18,16)(14,21,19,26)(15,24,25,28)(20,27,22,23)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,d^2=fi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,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, 6590997);