Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437770 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3006 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2876 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,12)(8,10)(9,11)(13,14,18,22,19,23,17,28)(15,24,25,27,26,20,16,21);(1,2)(3,4)(13,14)(15,24)(16,27)(17,22)(18,28)(19,23)(20,26)(21,25);(2,4)(5,7,9,8)(6,10,11,12)(13,15)(14,20)(16,17)(18,25)(19,26)(21,22)(23,24)(27,28);(2,4)(5,8,9,7)(6,10,11,12)(13,16,19,25)(14,21,23,27)(15,17,26,18)(20,22,24,28);(13,17,19,18)(14,22,23,28)(15,25,26,16)(20,27,24,21)Polycyclic presentation:
2^9|a^2=f,c^2=gi,d^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6359213);