Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667676 |
| 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 | #29622 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2254 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,9,12)(7,11,8,10)(13,14)(15,23)(16,20)(17,22)(18,21)(19,25)(24,28)(26,27);(1,2)(3,4)(13,15,18,25)(14,19,21,23)(16,24,17,26)(20,27,22,28);(2,4)(6,10)(7,8)(11,12)(13,16,18,17)(14,20,21,22)(15,24,25,26)(19,27,23,28);(14,21)(19,23)(20,22)(27,28);(2,4)(5,7,9,8)(6,11,12,10)(13,17,18,16)(14,22,21,20)(15,24,25,26)(19,27,23,28)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,g^c=gi71 conjugacy classes — view the full character table.
| maximal subgroup of | #72305239, #72305251, #72305265, … |
| maximal normal subgroup of | #72305239, #72305251, #72305265, … |
| maximal quotient of | #72305600, #72306069 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6582013);