Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6499792 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3562 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3176 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(13,14,19,22)(15,25,27,36)(16,30,32,39)(17,24,18,23)(20,29,37,28)(21,34,40,33)(26,38,31,35)(41,43,44,42);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,31)(17,29)(18,28)(19,27)(21,38)(22,37)(23,36)(24,25)(26,32)(30,41)(33,42)(34,43)(35,40)(39,44);(6,9)(7,11)(13,16)(14,21)(15,26)(17,33)(18,34)(19,32)(20,35)(22,40)(23,30)(24,39)(25,41)(27,31)(28,42)(29,43)(36,44)(37,38);(5,7,8,11)(6,10,9,12)(14,22)(15,27)(16,32)(23,24)(28,29)(33,34)(35,38)(41,44);(6,9)(10,12)(13,17,19,18)(14,23,22,24)(15,28,27,29)(16,33,32,34)(20,25,37,36)(21,30,40,39)(26,42,31,43)(35,41,38,44)Polycyclic presentation:
2^9|a^2=i,d^2=hi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6420038);