Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6668044 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2147 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)(7,10)(8,12)(9,11)(13,14)(15,18)(16,19)(17,20)(21,24)(22,26)(23,25)(27,28);(1,2)(3,4)(5,7,10,6)(8,12,9,11)(13,15)(14,18)(16,23)(17,22)(19,25)(20,26)(21,27)(24,28);(2,4)(14,19)(16,24)(18,25)(20,21)(23,28)(26,27);(5,8,10,9)(6,11,7,12)(15,22)(18,26)(23,28)(25,27);(2,4)(5,9,10,8)(6,11,7,12)(14,20)(18,26)(19,21)(25,27)Polycyclic presentation:
2^9|b^2=f,d^2=f,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72284323, #72284625, #72307763, … |
| maximal normal subgroup of | #72284323, #72284625, #72307763, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584264);