Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | #34977 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2203 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,19,20,18,17,15,22)(16,21,26,27,24,25,23,28);(5,7,8,12)(6,9,10,11)(13,15,18,19)(14,20,17,22)(16,23,24,26)(21,27,25,28);(2,4)(7,12)(9,11)(13,16)(14,21)(15,23)(17,25)(18,24)(19,26)(20,27)(22,28);(6,10)(9,11)(14,17)(16,24)(20,22)(23,26);(13,17,19,22,18,14,15,20)(16,25,26,28,24,21,23,27)Polycyclic presentation:
2^9|a^2=h,b^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=cfi,e^b=ei,d^c=di,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6284881);