Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2205 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6)(7,11)(8,9)(10,12)(13,14)(15,16)(17,27)(18,28)(19,22)(20,25)(21,23)(24,26);(2,3)(5,6)(7,10)(8,9)(11,12)(13,15,23,17,19,24,18,25)(14,20,28,26,22,27,21,16);(1,2,4,3)(6,9)(10,11)(13,16)(14,15)(17,28)(18,27)(19,26)(20,23)(21,25)(22,24);(1,3,4,2)(5,7,8,12)(6,10,9,11)(15,24)(16,26)(17,25)(20,27);(13,17,18,15,19,25,23,24)(14,16,21,27,22,26,28,20)Polycyclic presentation:
2^9|b^2=fi,c^2=hi,d^2=g,e^2=f,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72314158, #72314231 |
| maximal normal subgroup of | #72314158, #72314231 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7504193);