Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,16)(15,21,22,28)(17,25,27,20)(19,26,24,23);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(15,22)(19,24)(21,28)(23,26);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,24)(17,26)(18,22)(20,28)(21,25)(23,27);(2,4)(7,12)(9,11)(13,16,18,14)(15,19,22,24)(17,25,27,20)(21,26,28,23);(6,10)(9,11)(14,16)(19,24)(20,25)(21,28)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6961261);