Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #232351 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
1040 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6,9,7)(8,10,12,11)(13,14,18,22)(15,19,25,28)(16,20,17,21)(23,27,24,26);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,18,25)(14,19,22,28)(16,24,17,23)(20,26,21,27);(1,3,4,2)(6,10)(7,11)(8,12)(14,20)(15,23)(16,17)(21,22)(24,25)(26,27);(13,16,18,17)(14,20,22,21)(15,24,25,23)(19,26,28,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=e,d^2=hi,f^2=h,g^2=h,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=dhi,f^c=fh,g^c=gh68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 145666);