Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2498 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)(7,9)(8,10)(11,12)(13,14)(15,23)(16,19)(17,18)(20,22)(21,26)(24,25)(27,28);(5,6)(7,11)(8,10)(9,12)(13,15)(14,21)(16,23)(17,28)(18,26)(19,25)(20,27)(22,24);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16,20,17)(14,18,22,19)(15,23,27,28)(21,26,24,25);(7,12)(9,11)(13,17,20,16)(14,18,22,19)(15,24,27,21)(23,25,28,26);(6,10)(9,11)(13,18,20,19)(14,17,22,16)(15,25,27,26)(21,23,24,28)Polycyclic presentation:
2^9|c^2=i,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=ehi,f^b=fi,d^c=dh,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72283056, #72283209, #72283803, … |
| maximal normal subgroup of | #72283056, #72283209, #72283803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6901214);