Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3115 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2306 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(13,14,17,22)(15,19,18,27)(16,25,26,20)(21,23,28,24);(6,9)(10,12)(13,15,17,18)(14,19,22,27)(16,24,26,23)(20,28,25,21);(1,2,3,4)(13,16)(14,20)(15,23)(17,26)(18,24)(19,21)(22,25)(27,28);(13,16)(14,21)(15,24)(17,26)(18,23)(19,20)(22,28)(25,27);(5,7,8,11)(6,10,9,12)(13,15,17,18)(14,19,22,27)(16,24,26,23)(20,28,25,21)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dh,h^a=hi,c^b=ci,e^c=ei,h^d=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537456);