Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | #35168 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2196 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)(15,19)(16,18)(17,20)(21,26)(22,23)(24,28)(25,27);(5,7,9,6)(8,11,12,10)(13,15,17,24)(14,19,20,28)(16,23,26,27)(18,22,21,25);(2,4)(5,8,9,12)(6,10,7,11)(14,20)(15,22)(18,21)(19,27)(23,28)(24,25);(2,4)(6,7)(10,11)(13,16)(14,21)(15,23)(17,26)(18,20)(19,25)(22,28)(24,27);(14,20)(16,26)(19,28)(23,27)Polycyclic presentation:
2^9|a^2=gi,b^2=fi,c^2=f,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 | #72284115, #72284652, #72284692 |
| maximal normal subgroup of | #72284115, #72284652, #72284692 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410748);