Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6438975 |
| 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 | #23421 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2499 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)(13,14,20,25)(15,26,29,22)(16,21,28,36)(17,23,32,40)(18,24,19,34)(27,41,42,37)(30,35,43,44)(31,38,33,39);(1,2,3,4)(7,8)(10,12)(13,15,18,28,20,29,19,16)(14,21,34,22,25,36,24,26)(17,27,33,43,32,42,31,30)(23,35,38,37,40,44,39,41);(5,7,9,8)(6,10,11,12)(13,16,19,29,20,28,18,15)(14,22,24,21,25,26,34,36)(17,30,31,42,32,43,33,27)(23,37,39,35,40,41,38,44);(5,8,9,7)(6,10,11,12)(13,17)(14,23)(15,27)(16,30)(18,33)(19,31)(20,32)(21,35)(22,37)(24,39)(25,40)(26,41)(28,43)(29,42)(34,38)(36,44);(13,18,20,19)(14,24,25,34)(15,28,29,16)(17,31,32,33)(21,26,36,22)(23,38,40,39)(27,30,42,43)(35,37,44,41)Polycyclic presentation:
2^9|a^2=i,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=dh,e^d=ei86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370409);