Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806575 |
| 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 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2242 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)(7,10)(8,9)(11,12)(15,24)(17,25)(19,27)(21,28);(6,9)(7,11)(13,14)(15,21)(16,20)(17,27)(18,22)(19,25)(23,26)(24,28);(1,2,3,4)(13,15,16,25,18,24,26,17)(14,19,23,21,22,27,20,28);(6,9)(10,12)(13,16,18,26)(14,20,22,23)(15,25,24,17)(19,28,27,21);(5,7,8,11)(6,10,9,12)(13,17,26,24,18,25,16,15)(14,21,20,19,22,28,23,27)Polycyclic presentation:
2^9|c^2=ghi,d^2=i,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723064);