Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2084 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)(7,9)(8,10)(11,12)(13,14,17,20,18,21,16,24)(15,19,22,27,23,28,25,26);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,15,18,23)(14,19,21,28)(16,25,17,22)(20,27,24,26);(2,4)(5,8,9,12)(6,10,7,11)(13,14)(15,19)(16,20)(17,24)(18,21)(22,26)(23,28)(25,27);(6,7)(10,11)(13,14)(15,19)(16,20)(17,24)(18,21)(22,26)(23,28)(25,27);(13,16,18,17)(14,20,21,24)(15,22,23,25)(19,26,28,27)Polycyclic presentation:
2^9|a^2=g,b^2=fi,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410824);