Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438997 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
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 | |
2467 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14)(15,27)(16,32)(17,23)(18,24)(19,25)(20,26)(21,29)(22,33)(28,41)(30,40)(31,39)(34,38)(35,42)(36,43)(37,44);(1,2,3,4)(7,8)(10,12)(13,15,19,30,20,31,18,29)(14,21,24,39,26,40,25,27)(16,17,33,35,34,37,28,36)(22,23,32,43,41,44,38,42);(5,7,9,8)(6,10,11,12)(13,16,19,33,20,34,18,28)(14,22,24,32,26,41,25,38)(15,17,30,35,31,37,29,36)(21,23,39,43,40,44,27,42);(5,8,9,7)(6,10,11,12)(13,17)(14,23)(15,28)(16,30)(18,36)(19,35)(20,37)(21,38)(22,39)(24,43)(25,42)(26,44)(27,41)(29,34)(31,33)(32,40);(13,18,20,19)(14,24,26,25)(15,29,31,30)(16,33,34,28)(17,35,37,36)(21,39,40,27)(22,38,41,32)(23,42,44,43)Polycyclic presentation:
2^9|b^2=fi,c^2=g,d^2=h,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^c=ei,e^d=ei80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370428);