Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667836 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2208 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)(7,11)(8,10)(9,12)(13,14,18,20)(15,24,26,21)(16,27,19,22)(17,23,25,28);(5,7,8,12)(6,9,10,11)(14,20)(15,25)(16,19)(17,26)(21,28)(23,24);(1,2,3,4)(13,15)(14,21)(16,25)(17,19)(18,26)(20,24)(22,28)(23,27);(6,10)(9,11)(13,16,18,19)(14,22,20,27)(15,25,26,17)(21,28,24,23);(7,12)(9,11)(13,17,18,25)(14,23,20,28)(15,19,26,16)(21,27,24,22)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=gi,d^2=i,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,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6590051);