Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628847 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2152 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)(5,6,8,10)(7,11,12,9)(13,14,18,23)(15,24,19,20)(16,26,17,21)(22,27,28,25);(5,7,8,12)(6,9,10,11)(13,15,18,19)(14,20,23,24)(16,27,17,25)(21,28,26,22);(1,2,3,4)(13,16,18,17)(14,21,23,26)(15,25,19,27)(20,22,24,28);(13,17,18,16)(14,22,23,28)(15,25,19,27)(20,21,24,26);(14,23)(20,24)(21,26)(22,28)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=g,d^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ei,h^a=hi,c^b=ci,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6537300);