Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62307 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
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 | C2×#3542 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3444 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,25)(16,27)(17,22)(18,24)(19,23)(20,21)(26,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,23)(17,27)(18,21)(19,26)(22,28)(24,25);(2,4)(13,16,19,20,18,28,17,25)(14,21,22,27,24,15,23,26);(7,12)(9,11)(13,17,18,19)(14,22,24,23)(15,26,21,27)(16,25,28,20);(6,10)(9,11)(13,17,18,19)(14,23,24,22)(15,26,21,27)(16,20,28,25)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,e^c=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300687);