Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #162342 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #35001 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1644 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,18,22,25)(16,19)(17,20)(21,24,27,28)(23,26);(5,7,8,6,10,12,9,11)(13,15,17,22)(14,18)(16,21,23,27)(19,24)(20,25)(26,28);(1,2,3,4)(13,16)(14,19)(15,21)(17,23)(18,24)(20,26)(22,27)(25,28);(5,8,10,9)(6,7,11,12)(15,22)(16,23)(18,25)(19,26)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=f,d^2=hi,e^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=ei,g^b=gi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 111844);