Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2417 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(7,11)(10,12)(13,14,18,21,19,22,16,17)(15,23,26,27,25,28,24,20);(6,9)(7,11)(13,15)(14,20)(16,26)(17,23)(18,24)(19,25)(21,28)(22,27);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,19,18)(14,21,22,17)(15,24,25,26)(20,23,27,28);(6,9)(10,12)(13,17,16,22,19,21,18,14)(15,20,24,28,25,27,26,23);(5,7,8,11)(6,10,9,12)(14,22)(15,25)(17,21)(24,26)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72308248, #72308285, #72308361 |
| maximal normal subgroup of | #72308248, #72308285, #72308361 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053234);