Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667772 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2269 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(14,20)(15,22)(21,28)(24,25);(5,7)(6,9)(8,12)(10,11)(13,14,18,20)(15,23,22,26)(16,21,19,28)(17,25,27,24);(1,2,3,4)(13,15,18,22)(14,17,20,27)(16,24,19,25)(21,26,28,23);(6,10)(9,11)(13,16,18,19)(14,21,20,28)(15,24,22,25)(17,26,27,23);(7,12)(9,11)(13,17,18,27)(14,15,20,22)(16,23,19,26)(21,25,28,24)Polycyclic presentation:
2^9|b^2=i,c^2=g,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6589907);