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 | |
2062 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)(15,24)(16,23)(17,22)(18,21)(19,26)(20,25)(27,28);(1,2)(3,4)(5,7,10,6)(8,12,9,11)(13,15,17,26)(14,19,22,24)(16,27,18,25)(20,21,28,23);(2,4)(13,15)(14,20)(16,25)(17,26)(18,27)(19,23)(21,24)(22,28);(5,8,10,9)(6,11,7,12)(13,16,17,18)(14,21,22,23)(15,25,26,27)(19,20,24,28);(2,4)(5,9,10,8)(6,11,7,12)(14,22)(19,24)(20,28)(21,23)Polycyclic presentation:
2^9|b^2=f,d^2=f,e^2=fi,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 | #72308006, #72308285, #72308409 |
| maximal normal subgroup of | #72308006, #72308285, #72308409 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6584265);