Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61643 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#540 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#533 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2832 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,31,26,30,25,35)(23,32,33,29,24,28,34,36);(5,7,9,15)(6,10,12,13)(8,16,17,14)(11,19,20,18)(21,23,26,24)(22,28,30,32)(25,34,27,33)(29,31,36,35);(1,2,3,4)(21,24,26,23)(22,29,30,36)(25,34,27,33)(28,31,32,35);(5,8,9,17)(6,11,12,20)(7,14,15,16)(10,18,13,19)(22,30)(28,32)(29,36)(31,35);(6,12)(10,13)(11,20)(18,19)(21,25,26,27)(22,31,30,35)(23,33,24,34)(28,29,32,36)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=gh,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,d^b=dfi,e^b=ei,g^c=gi,h^c=hi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300838);