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 | |
2142 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)(6,9)(10,11)(13,14,17,20)(15,16,19,18)(21,22,26,27)(23,31,32,28)(24,34,35,29)(25,33,36,30);(5,6)(7,11)(8,9)(10,12)(13,15,17,19)(14,18,20,16)(22,27)(23,32)(24,35)(25,36);(6,9)(10,11)(13,16,17,18)(14,19,20,15)(21,23,26,32)(22,28,27,31)(24,30,35,33)(25,34,36,29);(1,2,3,4)(5,6)(7,10)(8,9)(11,12)(13,16)(14,15)(17,18)(19,20)(21,24,26,35)(22,29,27,34)(23,33,32,30)(25,28,36,31);(5,7)(6,10)(8,12)(9,11)(13,15)(14,16)(17,19)(18,20)(21,25,26,36)(22,30,27,33)(23,34,32,29)(24,28,35,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=fgh,d^2=fgi,e^2=g,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, 1652922);