Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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×#3214 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2103 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,10)(9,12)(15,22)(16,17)(19,27)(20,25)(23,28)(24,26);(2,3)(5,7,8,12)(6,9,10,11)(13,14,18,21)(15,23,24,19)(16,25,17,20)(22,28,26,27);(1,2,4,3)(7,12)(9,11)(13,15,18,24)(14,19,21,23)(16,22,17,26)(20,28,25,27);(6,10)(9,11)(13,14)(15,19)(16,25)(17,20)(18,21)(22,27)(23,24)(26,28);(1,3,4,2)(13,16,18,17)(14,20,21,25)(15,22,24,26)(19,28,23,27)Polycyclic presentation:
2^9|b^2=fi,c^2=fh,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6717763);