Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500701 |
| 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 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2082 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)(5,6)(7,9)(8,10)(11,12)(15,23)(19,27)(20,28)(24,25);(5,7,9,6)(8,11,12,10)(13,14,17,22)(15,20,23,28)(16,21,18,26)(19,25,27,24);(2,4)(5,8,9,12)(6,10,7,11)(13,15)(14,19)(16,24)(17,23)(18,25)(20,26)(21,28)(22,27);(2,4)(6,7)(10,11)(13,15)(14,20)(16,25)(17,23)(18,24)(19,26)(21,27)(22,28);(13,16,17,18)(14,21,22,26)(15,24,23,25)(19,28,27,20)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410888);