Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| automorphism | Aut(G) order 1048576 |
| 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 | #16500 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2385 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)(7,11)(8,9)(10,12)(13,14,18,16)(15,20,17,19)(21,22,26,27)(23,28,29,24)(25,30,34,35)(31,33,36,32);(5,6)(7,11)(8,9)(10,12)(13,14,18,16)(15,19,17,20)(22,27)(23,29)(30,35)(31,36);(6,9)(10,11)(13,15,18,17)(14,19,16,20)(21,23)(22,28)(24,27)(25,31)(26,29)(30,33)(32,35)(34,36);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,16,18,14)(15,19,17,20)(21,24,26,28)(22,29,27,23)(25,33,34,32)(30,31,35,36);(5,7)(6,10)(8,12)(9,11)(13,17,18,15)(14,19,16,20)(21,25)(22,30)(23,31)(24,32)(26,34)(27,35)(28,33)(29,36)Polycyclic presentation:
2^9|a^2=gh,b^2=fgh,c^2=fgh,d^2=gh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5545689);