Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806903 |
| 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 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#121 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1787 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)(7,11)(8,10)(9,12)(13,14,18,22)(15,19,16,20)(17,21,24,27)(23,26,25,28);(5,7,8,12)(6,9,10,11)(15,23)(16,25)(17,24)(19,26)(20,28)(21,27);(1,2,3,4)(6,10)(7,12)(13,15,24,25,18,16,17,23)(14,19,27,28,22,20,21,26);(6,10)(9,11)(15,16)(19,20)(23,25)(26,28);(13,16,24,23,18,15,17,25)(14,20,27,26,22,19,21,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=ghi,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6723715);