Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470003 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3028 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)(7,11)(10,12)(13,14,18,24,19,23,28,26)(15,25,21,17,22,16,27,20);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,24)(17,23)(18,27)(19,22)(21,28)(25,26);(2,4)(6,9)(10,12)(13,16)(14,21)(15,26)(17,28)(18,20)(19,25)(22,24)(23,27);(6,9)(10,12)(13,17,19,20)(14,22,23,15)(16,28,25,18)(21,24,27,26);(5,7,8,11)(6,10,9,12)(14,23)(16,25)(17,20)(24,26)Polycyclic presentation:
2^9|a^2=f,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6398451);