Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #188176 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C24 | C23×C4 |
| Frattini, prime radical, cosocle | C25 | C24 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4⋊C4 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1672 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,10)(8,12)(9,11)(13,14)(15,23,26,19)(16,20)(17,27,24,21)(18,22)(25,28);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,24)(17,25)(18,26)(20,27)(21,28)(22,23);(2,4)(6,11)(10,12)(13,16,18,25)(14,20,22,28)(15,24,26,17)(19,27,23,21);(5,8,9,7)(6,12,11,10)(13,17,18,24)(14,21,22,27)(15,25,26,16)(19,28,23,20)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=i,d^2=fi,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,h^b=hi,h^d=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268469);