Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| 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 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2779 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)(7,9)(8,10)(11,12)(13,14)(15,24)(16,27)(17,23)(18,26)(19,21)(20,22)(25,28);(5,7)(6,9)(8,11)(10,12)(13,15,18,22,19,25,17,27)(14,20,23,24,21,16,26,28);(1,2,3,4)(7,11)(9,12)(14,21)(15,25)(22,27)(23,26);(6,10)(9,12)(13,16)(14,22)(15,26)(17,24)(18,28)(19,20)(21,27)(23,25);(13,17,19,18)(14,23,21,26)(15,27,25,22)(16,28,20,24)Polycyclic presentation:
2^9|b^2=f,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6341004);