Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3111 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)(13,14,19,27,18,21,17,22)(15,23,26,16,20,28,24,25);(5,6)(7,10)(8,9)(11,12)(13,15,18,20)(14,16,21,25)(17,26,19,24)(22,28,27,23);(2,4)(6,9)(10,12)(13,16)(14,20)(15,21)(17,23)(18,25)(19,28)(22,26)(24,27);(5,7)(6,10)(8,11)(9,12)(14,21)(16,25)(22,27)(23,28);(7,11)(10,12)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,f^b=fi,d^c=di,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6449781);