Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| 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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3031 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,23)(16,21)(17,20)(18,22)(19,25)(24,27)(26,28);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,14)(15,24)(16,20)(17,21)(18,22)(19,26)(23,27)(25,28);(7,11)(9,12)(13,15)(14,19)(16,25)(17,27)(18,26)(20,23)(21,28)(22,24);(6,10)(9,12)(13,16,18,17)(14,20,22,21)(15,25,26,27)(19,23,24,28);(13,16,18,17)(14,21,22,20)(15,25,26,27)(19,28,24,23)Polycyclic presentation:
2^9|b^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72312897, #72312907, #72312961 |
| maximal normal subgroup of | #72312897, #72312907, #72312961 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340198);