Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573477 |
| 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×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2959 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,12)(8,10)(9,11)(13,14,19,18)(15,20,17,22)(16,21,26,28)(23,25,27,24);(5,7,9,8)(6,10,11,12)(13,15,19,17)(14,20,18,22)(16,24,26,25)(21,23,28,27);(2,4)(13,16,19,26)(14,21,18,28)(15,23,17,27)(20,25,22,24);(5,8,9,7)(6,10,11,12)(13,17,19,15)(14,22,18,20)(16,25,26,24)(21,27,28,23);(14,18)(15,22)(17,20)(21,28)(23,24)(25,27)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=i,d^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6483804);