Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 |
| prime radical | #45263 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2234 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)(5,6)(7,11)(8,10)(9,12)(13,14,18,19)(15,20,16,23)(17,21,26,27)(22,28,24,25);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,18)(17,25)(19,23)(21,22)(24,27)(26,28);(2,4)(13,16,18,15)(14,20,19,23)(17,24,26,22)(21,28,27,25);(2,4)(5,7,9,8)(6,11,12,10)(13,17,18,26)(14,21,19,27)(15,22,16,24)(20,28,23,25);(5,8,9,7)(6,11,12,10)(15,16)(20,23)(22,24)(25,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7429681);