Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60753 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1824 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(6,9)(7,11)(13,14)(15,25)(16,20)(17,24)(18,23)(19,22)(21,27)(26,28);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,18)(17,27)(19,26)(21,22)(23,25)(24,28);(6,9)(10,12)(13,16,18,15)(14,21,23,28)(17,27,19,26)(20,24,25,22);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,22,23,24)(15,26,16,27)(20,28,25,21);(14,23)(20,25)(21,28)(22,24)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=cf,d^b=di,g^c=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282940);