Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2280 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,20)(16,23)(17,27)(18,21)(19,25)(22,24)(26,28);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,14)(15,23)(16,20)(17,22)(18,21)(19,26)(24,27)(25,28);(2,4)(13,15,18,26)(14,19,21,23)(16,24,25,17)(20,22,28,27);(6,10)(7,12)(13,16)(14,20)(15,24)(17,26)(18,25)(19,22)(21,28)(23,27);(14,21)(19,23)(20,28)(22,27)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,g^a=gi,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, 7408272);