Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2266 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)(7,11)(10,12)(13,14,20,26)(15,27,32,38)(16,22,17,23)(18,25,19,24)(21,30,39,31)(28,43,34,37)(29,41,44,36)(33,42,35,40);(6,9)(7,11)(13,15,19,31,20,32,18,30)(14,21,25,27,26,39,24,38)(16,28,33,44,17,34,35,29)(22,36,40,43,23,41,42,37);(2,4)(5,6)(7,10)(8,9)(11,12)(13,16,20,17)(14,22,26,23)(15,28,32,34)(18,35,19,33)(21,36,39,41)(24,42,25,40)(27,43,38,37)(29,31,44,30);(6,9)(10,12)(13,17,20,16)(14,23,26,22)(15,29,32,44)(18,35,19,33)(21,37,39,43)(24,42,25,40)(27,41,38,36)(28,31,34,30);(5,7,8,11)(6,10,9,12)(13,18,20,19)(14,24,26,25)(15,30,32,31)(16,33,17,35)(21,38,39,27)(22,40,23,42)(28,44,34,29)(36,43,41,37)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=i,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fi,c^b=ch,d^b=dfi,e^b=eh,d^c=dh,e^c=ei,f^d=fi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7051018);