Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #45136 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1833 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)(7,9)(8,10)(11,12)(13,14,18,24,19,25,20,26)(15,27,35,38,31,21,30,40)(16,23,34,22,17,33,32,39)(28,42,43,41,29,37,44,36);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15)(14,21)(16,28)(17,29)(18,35)(19,31)(20,30)(22,36)(23,37)(24,40)(25,27)(26,38)(32,44)(33,42)(34,43)(39,41);(5,7,8,12)(6,9,10,11)(13,16,19,17)(14,22,25,39)(15,28,31,29)(18,32,20,34)(21,36,27,41)(23,26,33,24)(30,43,35,44)(37,38,42,40);(1,3,4,2)(7,12)(9,11)(13,17,19,16)(14,23,25,33)(15,29,31,28)(18,32,20,34)(21,37,27,42)(22,26,39,24)(30,43,35,44)(36,38,41,40);(6,10)(9,11)(13,18,19,20)(14,24,25,26)(15,30,31,35)(16,32,17,34)(21,38,27,40)(22,23,39,33)(28,43,29,44)(36,42,41,37)Polycyclic presentation:
2^9|a^2=fg,c^2=hi,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dfi,e^b=ehi,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6971856);