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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2420 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)(5,6,8,9)(7,11,12,10)(13,14,18,20)(15,17,16,19)(21,22)(23,28)(24,32)(25,30)(26,27)(29,35)(31,34)(33,36);(6,9)(7,12)(13,15)(14,19)(16,18)(17,20)(22,27)(25,28)(29,32)(34,36);(6,9)(10,11)(14,20)(15,16)(21,23,26,30)(22,28,27,25)(24,33,35,31)(29,34,32,36);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,15)(14,17,20,19)(21,24,26,35)(22,29,27,32)(23,31,30,33)(25,34,28,36);(5,7)(6,10)(8,12)(9,11)(13,17)(14,15)(16,20)(18,19)(21,25,26,28)(22,30,27,23)(24,34,35,36)(29,33,32,31)Polycyclic presentation:
2^9|a^2=g,c^2=fg,d^2=ghi,e^2=fg,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, 1488691);