Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34737 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2499 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)(6,9)(10,12)(13,14,18,22,17,21,26,28)(15,23,24,16,20,19,27,25);(5,6)(7,10)(8,9)(11,12)(13,15,18,24,17,20,26,27)(14,19,22,25,21,23,28,16);(1,2,3,4)(5,7,8,11)(6,10,9,12)(13,16,17,25)(14,20,21,15)(18,19,26,23)(22,27,28,24);(6,9)(10,12)(14,21)(15,20)(22,28)(24,27);(15,20)(16,25)(19,23)(24,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fg,h^2=i,b^a=bf,c^a=cg,d^a=di,d^b=df,e^b=ei,e^c=ei116 conjugacy classes — view the full character table.
| maximal subgroup of | #72308260, #72308271, #72308325 |
| maximal normal subgroup of | #72308260, #72308271, #72308325 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6289762);