Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3020 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)(7,11)(8,10)(9,12)(13,14)(15,18)(16,20)(17,19)(21,25)(22,27)(23,26)(24,28);(5,7)(6,9)(8,12)(10,11)(13,15)(14,18)(16,21)(17,23)(19,26)(20,25)(22,24)(27,28);(2,4)(15,21)(16,24)(18,25)(20,28)(22,23)(26,27);(2,4)(6,10)(7,12)(13,14)(15,18)(16,20)(17,19)(21,25)(22,27)(23,26)(24,28);(14,19)(18,26)(20,28)(25,27)Polycyclic presentation:
2^9|a^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,h^b=hi,h^c=hi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72284037, #72284541, #72284692, … |
| maximal normal subgroup of | #72284037, #72284541, #72284692, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410329);