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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1860 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14,17,16)(15,18,20,19)(21,22,26,27)(23,28,32,34)(24,31,35,29)(25,30,36,33);(7,12)(9,11)(13,14,17,16)(15,19,20,18)(22,27)(25,36)(28,34)(29,31);(5,7)(6,9)(8,12)(10,11)(13,15,17,20)(14,18,16,19)(21,23)(22,28)(24,33)(25,31)(26,32)(27,34)(29,36)(30,35);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(21,24,26,35)(22,29,27,31)(23,30,32,33)(25,34,36,28);(5,6)(7,9)(8,10)(11,12)(13,16,17,14)(15,19,20,18)(21,25)(22,30)(23,31)(24,34)(26,36)(27,33)(28,35)(29,32)Polycyclic presentation:
2^9|a^2=fh,b^2=gh,c^2=gh,d^2=gi,e^2=gh,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, 1588681);