Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518964 |
| 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 | #45004 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ 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:
(5,6,9,11)(7,10,12,8)(13,14,20,25)(15,16,22,24)(17,23,19,28)(18,21,26,27);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,19,18,20,22,17,26)(14,21,23,16,25,27,28,24);(6,11)(7,12)(13,16)(14,22)(15,25)(17,21)(18,28)(19,27)(20,24)(23,26);(2,4)(5,8,9,10)(6,12,11,7)(13,17,20,19)(14,23,25,28)(15,26,22,18)(16,27,24,21);(13,18,17,15,20,26,19,22)(14,24,28,27,25,16,23,21)Polycyclic presentation:
2^9|a^2=gi,b^2=f,d^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72281938 |
| maximal normal subgroup of | #72281938 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7464380);