Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1004 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6)(7,11)(8,13)(9,18)(10,14)(12,15)(16,20)(17,19)(21,22,24,28,25,27,31,34)(23,26,30,32,29,33,35,36);(1,2,3,4)(5,7)(6,11)(8,17)(9,15)(10,16)(12,18)(13,19)(14,20);(5,7)(6,12)(8,15)(9,17)(10,16)(11,18)(13,20)(14,19)(21,23)(22,26)(24,30)(25,29)(27,33)(28,32)(31,35)(34,36);(5,8,10,9)(6,13,14,18)(7,15,16,17)(11,12,20,19)(22,27)(23,29)(28,34)(30,35)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,f^a=fh,d^b=dh,d^c=di,f^c=fh92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96688);