Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| socle | C23 | C23×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2298 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,8,11,10)(13,14,17,21,18,22,16,20)(15,23,24,27,25,28,26,19);(1,2)(3,4)(5,7,9,11)(6,10,12,8);(2,4)(5,8,9,10)(6,11,12,7)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,28)(22,27);(6,12)(7,11)(13,15)(14,19)(16,24)(17,26)(18,25)(20,23)(21,28)(22,27);(13,16,18,17)(14,20,22,21)(15,24,25,26)(19,23,27,28)Polycyclic presentation:
2^9|a^2=fg,b^2=f,c^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,g^c=gi,g^d=gi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410981);