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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,16)(15,19,17,20)(21,22)(23,29)(24,32)(25,26)(27,31)(28,33)(30,35)(34,36);(5,6,8,10)(7,9,11,12)(13,15,18,17)(14,19,16,20)(22,25)(24,33)(27,29)(30,34);(5,7,8,11)(6,9,10,12)(13,16,18,14)(15,19,17,20)(21,23)(22,27)(24,34)(25,29)(26,31)(28,36)(30,33)(32,35);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,17,18,15)(14,19,16,20)(21,24)(22,28)(23,30)(25,32)(26,33)(27,35)(29,36)(31,34);(5,6)(7,9)(8,10)(11,12)(21,25,26,22)(23,29,31,27)(24,32,33,28)(30,36,34,35)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fgh,d^2=gi,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, 1554241);