Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60460 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27×C4 |
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 | C24 | C22×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2070 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,22)(16,21)(17,23)(18,24)(19,25)(20,26)(27,37)(28,36)(29,35)(30,34)(31,38)(32,39)(33,40)(41,44)(42,43);(6,9)(10,12)(13,15,19,16,20,29,18,28)(14,21,24,22,26,36,25,35)(17,27,31,30,33,42,32,41)(23,34,39,37,40,44,38,43);(5,7,8,11)(6,10,9,12)(13,16,19,29,20,28,18,15)(14,22,24,36,26,35,25,21)(17,30,31,42,33,41,32,27)(23,37,39,44,40,43,38,34);(1,3,4,2)(13,17,20,33)(14,23,26,40)(15,27,29,42)(16,30,28,41)(18,32,19,31)(21,34,36,44)(22,37,35,43)(24,39,25,38);(13,18,20,19)(14,24,26,25)(15,28,29,16)(17,31,33,32)(21,22,36,35)(23,38,40,39)(27,30,42,41)(34,43,44,37)Polycyclic presentation:
2^9|a^2=h,b^2=gi,c^2=f,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ci,e^d=ei116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6280742);