Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23439 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1126 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,20,18,21,16,22)(15,19,25,28,23,27,24,26);(6,9)(10,12)(13,15,16,24,18,23,17,25)(14,19,22,26,21,27,20,28);(14,20)(15,23)(16,17)(19,26)(21,22)(27,28);(5,7,8,11)(6,10,9,12)(14,21)(19,27)(20,22)(26,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,f^c=fi,h^c=hi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72321621 |
| maximal normal subgroup of | #72321621 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 265612);