Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438949 |
| 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 | C22×#135 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,11)(8,10)(9,12)(13,14,19,23)(15,20,26,36)(16,27,28,41)(17,22,18,32)(21,29,37,30)(24,38,39,44)(25,34,31,35)(33,42,43,40);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,15)(14,20)(16,24)(17,31)(18,25)(19,26)(21,33)(22,35)(23,36)(27,38)(28,39)(29,42)(30,40)(32,34)(37,43)(41,44);(2,4)(5,7,9,8)(6,11,12,10)(13,16,18,30,19,28,17,29)(14,21,22,27,23,37,32,41)(15,24,25,40,26,39,31,42)(20,33,35,38,36,43,34,44);(2,4)(5,8,9,7)(6,10,12,11)(16,28)(21,37)(24,39)(27,41)(29,30)(33,43)(38,44)(40,42);(13,17,19,18)(14,22,23,32)(15,25,26,31)(16,29,28,30)(20,34,36,35)(21,27,37,41)(24,40,39,42)(33,44,43,38)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fg,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6351717);