Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4330 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,25,28,36)(16,30,33,37)(17,22,18,23)(20,27,35,29)(21,32,38,31)(26,40,42,34)(39,41,44,43);(2,4)(5,7)(6,9)(8,11)(10,12)(13,15)(14,20)(16,26)(17,27)(18,29)(19,28)(21,34)(22,25)(23,36)(24,35)(30,44)(31,41)(32,43)(33,42)(37,39)(38,40);(7,11)(9,12)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,34)(22,37)(23,30)(24,38)(25,39)(27,41)(28,42)(29,43)(35,40)(36,44);(6,10)(9,12)(13,17,19,18)(14,22,24,23)(15,27,28,29)(16,31,33,32)(20,25,35,36)(21,37,38,30)(26,41,42,43)(34,39,40,44);(15,28)(20,35)(25,36)(26,42)(27,29)(34,40)(39,44)(41,43)Polycyclic presentation:
2^9|a^2=i,d^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,f^b=fi,g^b=gi,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, 6340127);