Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499780 |
| 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 | #34977 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,28)(25,27);(1,2,4,3)(5,6)(7,10)(8,9)(11,12);(1,3,4,2)(6,9)(7,11)(13,15)(14,19)(16,24)(17,23)(18,25)(20,27)(21,28)(22,26);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,28,27);(6,9)(10,12)(13,16,18,17)(14,20,21,26)(15,24,25,23)(19,27,28,22)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,d^c=dhi,g^c=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421353);