Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346398 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3537 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2882 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)(9,12)(13,14,18,22)(15,20,24,27)(16,21,17,26)(19,25,28,23);(5,6)(7,9)(8,10)(11,12)(13,15)(14,19)(16,23)(17,25)(18,24)(20,26)(21,27)(22,28);(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,25)(17,23)(18,24)(19,26)(21,28)(22,27);(2,4)(13,16,18,17)(14,21,22,26)(15,23,24,25)(19,27,28,20)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fi,d^c=dh,e^c=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253608);