Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| 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 | |
2031 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)(13,14,19,22,18,24,17,23)(15,20,27,28,16,26,25,21);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15,18,16)(14,20,24,26)(17,25,19,27)(21,22,28,23);(2,4)(13,16,18,15)(14,21,24,28)(17,25,19,27)(20,22,26,23);(6,11)(7,12)(14,22)(17,19)(20,28)(21,26)(23,24)(25,27);(5,8,9,10)(6,7,11,12)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,g^c=gi,g^d=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7411174);