Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500661 |
| 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 | C22×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2211 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)(13,14)(15,23)(16,21)(17,20)(18,22)(19,25)(24,28)(26,27);(5,7,9,6)(8,11,12,10)(15,24)(16,18)(19,27)(21,22)(23,28)(25,26);(2,4)(5,8,9,12)(6,10,7,11)(13,15)(14,19)(16,24)(17,25)(18,26)(20,23)(21,27)(22,28);(2,4)(6,7)(10,11)(14,20)(19,23)(21,22)(27,28);(13,16,17,18)(14,21,20,22)(15,24,25,26)(19,27,23,28)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^b=ei,h^b=hi,h^c=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410879);