Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499804 |
| 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 | C22×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2283 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2262 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(13,14,17,20)(15,18,22,27)(16,19,24,21)(23,25,28,26);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,23)(17,22)(19,25)(20,27)(21,26)(24,28);(6,9)(7,11)(14,19)(16,24)(18,25)(20,21)(23,28)(26,27);(5,7,8,11)(6,10,9,12)(14,20)(15,22)(19,21)(23,28);(6,9)(10,12)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72315848, #72315902, #72316159, … |
| maximal normal subgroup of | #72315848, #72315902, #72316159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420621);