Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573591 |
| 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×#3355 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1823 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)(13,14,20,28,19,24,18,23)(15,25,27,21,17,22,16,26);(5,6,8,9)(7,10,11,12)(13,15,19,17)(14,21,24,26)(16,18,27,20)(22,28,25,23);(1,2,3,4)(6,9)(7,11)(13,16)(14,21)(15,20)(17,18)(19,27)(22,28)(23,25)(24,26);(5,7)(6,10)(8,11)(9,12)(13,17,19,15)(14,22,24,25)(16,18,27,20)(21,28,26,23);(6,9)(7,11)(15,17)(16,27)(21,26)(22,25)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=gi,d^2=i,f^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=eh83 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6513874);