Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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×#3539 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3276 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,23,19,24,17,22)(15,20,26,34,27,35,28,36)(16,29,30,21,32,37,31,38)(25,39,42,33,41,44,40,43);(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,25)(17,28)(18,26)(19,27)(21,33)(22,36)(23,34)(24,35)(29,39)(30,42)(31,40)(32,41)(37,44)(38,43);(1,2,3,4)(13,16)(14,21)(15,25)(17,30)(18,31)(19,32)(20,33)(22,37)(23,29)(24,38)(26,40)(27,41)(28,42)(34,39)(35,43)(36,44);(7,12)(9,11)(13,17,19,18)(14,22,24,23)(15,26,27,28)(16,30,32,31)(20,34,35,36)(21,37,38,29)(25,40,41,42)(33,39,43,44);(6,10)(9,11)(15,27)(20,35)(25,41)(26,28)(33,43)(34,36)(39,44)(40,42)Polycyclic presentation:
2^9|a^2=h,c^2=gh,d^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,d^b=dfi,e^b=ei,g^c=gi,h^c=hi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6300614);