Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
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×#3207 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2152 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)(5,6)(7,11)(8,10)(9,12)(14,19)(15,23)(16,18)(20,27)(24,25)(26,28);(5,7,8,12)(6,9,10,11)(13,14)(15,20)(16,21)(17,19)(18,22)(23,28)(24,26)(25,27);(7,12)(9,11)(13,15)(14,20)(16,23)(17,24)(18,25)(19,26)(21,28)(22,27);(6,10)(9,11)(13,16,17,18)(14,21,19,22)(15,23,24,25)(20,28,26,27);(1,3,4,2)(14,19)(20,26)(21,22)(27,28)Polycyclic presentation:
2^9|a^2=h,b^2=fi,d^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=cfi,e^b=ei,g^c=gi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6284957);