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 |
| prime radical | #23391 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2881 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)(22,26)(23,27)(24,28);(5,7)(6,9)(8,12)(10,11)(15,21)(16,23)(18,25)(20,27)(22,24)(26,28);(2,4)(13,15,17,22)(14,18,19,26)(16,24,23,21)(20,28,27,25);(2,4)(6,10)(7,12);(14,19)(18,26)(20,27)(25,28)Polycyclic presentation:
2^9|a^2=g,c^2=i,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=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72308185, #72308227, #72308276 |
| maximal normal subgroup of | #72308185, #72308227, #72308276 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410311);