Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500120 |
| 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 |
| prime radical | C2×#3540 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3222 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)(13,14,19,20,18,22,17,21)(15,23,24,16,26,27,25,28);(5,6)(7,11)(8,10)(9,12)(13,15)(14,16)(17,24)(18,26)(19,25)(20,23)(21,27)(22,28);(2,4)(5,7)(6,9)(8,12)(10,11)(13,16)(14,15)(17,23)(18,28)(19,27)(20,24)(21,25)(22,26);(7,12)(9,11)(13,17,18,19)(14,20,22,21)(15,24,26,25)(16,23,28,27);(6,10)(9,11)(13,17,18,19)(14,21,22,20)(15,24,26,25)(16,27,28,23)Polycyclic presentation:
2^9|a^2=fi,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6407327);