Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
8878 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6,10,9)(7,8,12,11)(14,18)(16,23)(19,24)(22,27);(6,9)(7,12)(15,21)(19,24)(22,27)(26,28);(5,7,10,12)(6,11,9,8)(13,14,17,18)(15,19,21,24)(16,20,23,25)(22,26,27,28);(5,8)(6,7)(9,12)(10,11);(5,9,10,6)(7,8,12,11)(13,15)(14,19)(16,22)(17,21)(18,24)(20,26)(23,27)(25,28);(5,8)(6,7)(9,12)(10,11)(13,16)(14,20)(15,22)(17,23)(18,25)(19,26)(21,27)(24,28)Polycyclic presentation:
2^9|a^2=i,c^2=h,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh152 conjugacy classes — view the full character table.
| maximal subgroup of | #72315055 |
| maximal normal subgroup of | #72315055 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7533407);