Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4146 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(14,18)(15,19)(20,23)(21,26)(22,25)(27,28);(6,9)(10,12)(13,14,17,18)(15,19,22,25)(16,23,24,20)(21,27,28,26);(13,15,17,22)(14,19,18,25)(16,21,24,28)(20,26,23,27);(2,4)(13,16)(14,20)(15,21)(17,24)(18,23)(19,26)(22,28)(25,27);(5,7,8,11)(6,10,9,12)(16,24)(20,23)(21,28)(26,27)Polycyclic presentation:
2^9|b^2=i,c^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,d^b=di,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536544);