Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431710 |
| 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×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,10)(8,9)(11,12)(13,14,19,24,18,23,27,26)(15,25,21,16,28,20,17,22);(1,2)(3,4)(6,9)(10,12)(13,15)(14,20)(16,26)(17,27)(18,28)(19,21)(22,24)(23,25);(2,4)(13,16,18,22)(14,21,23,17)(15,26,28,24)(19,25,27,20);(5,7,8,11)(6,10,9,12)(13,17,18,21)(14,22,23,16)(15,27,28,19)(20,24,25,26);(14,23)(15,28)(17,21)(24,26)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=ei,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, 6339546);