Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| 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 | C2×#392 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2161 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(7,11)(10,12)(13,14,17,18)(15,19,20,16)(21,22,24,26)(23,27,28,25);(2,4)(5,6,9,8)(7,10,11,12)(14,18)(16,19)(21,22,24,26)(23,25,28,27);(1,2,3,4)(13,15,17,20)(14,19,18,16)(21,22)(23,25)(24,26)(27,28);(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,20)(18,19)(21,22)(23,25)(24,26)(27,28);(5,8,9,6)(7,12,11,10)(13,16)(14,15)(17,19)(18,20)(21,23,24,28)(22,25,26,27)Polycyclic presentation:
2^9|a^2=fg,b^2=fi,c^2=fgh,e^2=fi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1533732);