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 | C24×D4 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3402 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6)(7,11)(8,10)(9,12)(13,14,17,20)(15,18,22,27)(16,19,24,21)(23,25,28,26);(1,3,4,2)(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,23)(17,22)(19,25)(20,27)(21,26)(24,28);(7,12)(9,11)(14,19)(16,24)(18,25)(20,21)(23,28)(26,27);(6,10)(9,11);(14,20)(15,22)(19,21)(23,28)Polycyclic presentation:
2^9|a^2=h,b^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=cf,e^b=ei,g^c=gi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72308227, #72308252, #72308276, … |
| maximal normal subgroup of | #72308227, #72308252, #72308276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6282722);