Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437753 |
| 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 | C2×#532 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2326 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14,17,20,18,21,16,25)(15,19,24,27,22,26,23,28);(2,3)(6,9)(10,11)(13,14)(15,19)(16,20)(17,25)(18,21)(22,26)(23,27)(24,28);(1,2,4,3)(5,7,8,12)(6,10,9,11)(13,15)(14,19)(16,23)(17,24)(18,22)(20,27)(21,26)(25,28);(1,3,4,2)(15,22)(19,26)(23,24)(27,28);(13,16,18,17)(14,20,21,25)(15,23,22,24)(19,27,26,28)Polycyclic presentation:
2^9|a^2=fg,c^2=gh,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6373324);