Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535648 |
| 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 ◁ C42×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2522 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,10,11)(7,8,12,9)(13,14,17,20)(15,18,21,24)(16,19,23,26)(22,25,27,28);(5,7,9,6,10,12,8,11)(13,15)(14,18)(16,22)(17,21)(19,25)(20,24)(23,27)(26,28);(1,2,3,4)(6,11)(7,12)(15,21)(18,24)(22,27)(25,28);(5,8,10,9)(6,12,11,7)(13,16)(14,19)(15,22)(17,23)(18,25)(20,26)(21,27)(24,28);(5,8,10,9)(6,12,11,7)(16,23)(19,26)(22,27)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ei,e^d=ehi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6476781);