Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2021 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)(7,8)(9,11)(10,12)(13,14,18,19,17,22,15,20)(16,24,23,28,26,27,25,21);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,17,18)(14,19,22,20)(16,25,26,23)(21,24,28,27);(2,4)(14,20)(15,18)(19,22)(21,24)(23,25)(27,28);(6,11)(7,12)(13,16,17,26)(14,21,22,28)(15,23,18,25)(19,27,20,24);(5,8,9,10)(6,7,11,12)Polycyclic presentation:
2^9|a^2=g,b^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfgi,c^b=ch,d^b=df,g^c=gi,g^d=gi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411352);