Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535823 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2747 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)(15,23)(16,26)(17,21)(18,24)(19,22)(20,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,15,18,26,19,25,17,20)(14,16,21,23,22,27,24,28);(1,2,3,4)(6,9)(10,12)(13,16,19,27)(14,20,22,26)(15,24,25,21)(17,28,18,23);(5,7)(6,10)(8,11)(9,12)(15,25)(16,27)(20,26)(23,28);(7,11)(10,12)(13,17,19,18)(14,21,22,24)(15,20,25,26)(16,23,27,28)Polycyclic presentation:
2^9|b^2=f,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=ehi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307362, #72307590, #72307778, … |
| maximal normal subgroup of | #72307362, #72307590, #72307778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6465251);