Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #686200 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,9,8)(7,11,12,10)(13,14,17,18)(15,19,20,16);(13,15)(14,16)(17,20)(18,19)(22,25)(26,28);(1,2,3,4)(14,18)(15,20)(21,22)(23,26)(24,25)(27,28);(5,7,9,12)(6,10,8,11)(13,15)(14,19)(16,18)(17,20)(21,23,24,27)(22,26,25,28);(5,8,9,6)(7,11,12,10)(13,16,17,19)(14,15,18,20)Polycyclic presentation:
2^9|a^2=fi,c^2=fg,d^2=fhi,e^2=fi,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72336599, #72336613, #72336740 |
| maximal normal subgroup of | #72336599, #72336613, #72336740 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 665713);