Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1933 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)(14,20)(15,22)(16,27)(25,26);(1,2,3,4)(13,14)(15,23)(16,17)(18,20)(19,27)(21,26)(22,24)(25,28);(6,9)(10,12)(13,15,17,25,18,22,19,26)(14,21,27,24,20,28,16,23);(13,16,17,14,18,27,19,20)(15,24,25,28,22,23,26,21);(5,7,8,11)(6,10,9,12)(13,17,18,19)(14,16,20,27)(15,25,22,26)(21,23,28,24)Polycyclic presentation:
2^9|b^2=fi,c^2=hi,d^2=hi,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dh,h^b=hi,d^c=di,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6375111);