Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500416 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(13,14,19,22)(15,20,24,27)(16,17,21,18)(23,25,28,26);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,23)(17,25)(18,26)(19,24)(21,28)(22,27);(6,9)(7,11)(13,16,18,14,19,21,17,22)(15,23,26,20,24,28,25,27);(5,7,8,11)(6,10,9,12)(15,24)(20,27)(23,28)(25,26);(6,9)(10,12)(13,17,19,18)(14,21,22,16)(15,25,24,26)(20,28,27,23)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6426817);