Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6535815 |
| 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 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2552 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(13,14,19,23)(15,24,26,39)(16,21,29,35)(17,22,18,32)(20,27,34,28)(25,38,41,43)(30,36,31,37)(33,42,44,40);(5,6)(7,10)(8,9)(11,12)(13,15,18,28,19,26,17,27)(14,20,22,24,23,34,32,39)(16,25,30,40,29,41,31,42)(21,33,37,38,35,44,36,43);(1,2,3,4)(6,9)(10,12)(13,16,19,29)(14,21,23,35)(15,25,26,41)(17,31,18,30)(20,33,34,44)(22,37,32,36)(24,38,39,43)(27,42,28,40);(5,7)(6,10)(8,11)(9,12)(15,26)(16,29)(20,34)(21,35)(24,39)(27,28)(30,31)(36,37);(7,11)(10,12)(13,17,19,18)(14,22,23,32)(15,27,26,28)(16,30,29,31)(20,24,34,39)(21,36,35,37)(25,40,41,42)(33,43,44,38)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,d^c=di,e^c=ei,e^d=eh74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6465314);