Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437521 |
| 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×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3138 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)(7,10)(8,9)(11,12)(13,14,20,24)(15,18,23,19)(16,26,28,21)(17,22,27,25);(1,2,3,4)(13,15,19,14,20,23,18,24)(16,26,27,22,28,21,17,25);(6,9)(10,12)(13,16)(14,21)(15,25)(17,19)(18,27)(20,28)(22,23)(24,26);(13,17,20,27)(14,22,24,25)(15,21,23,26)(16,19,28,18);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,15,24,23)(16,27,28,17)(21,25,26,22)Polycyclic presentation:
2^9|a^2=i,b^2=f,d^2=i,e^2=g,f^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,h^a=hi,c^b=ch,d^b=dh,f^c=fi,h^c=hi,f^d=fi,h^d=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361895);