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 | C2×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3059 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)(7,9)(8,10)(11,12)(13,14,20,26)(15,27,31,38)(16,32,35,41)(17,23,33,40)(18,24,19,25)(21,30,39,29)(22,34,42,28)(36,43,37,44);(1,2,3,4)(5,7)(6,9)(8,11)(10,12)(13,15,18,29,20,31,19,30)(14,21,25,27,26,39,24,38)(16,33,34,36,35,17,28,37)(22,40,41,44,42,23,32,43);(6,10)(7,11)(13,16,18,34,20,35,19,28)(14,22,25,41,26,42,24,32)(15,17,29,37,31,33,30,36)(21,23,27,43,39,40,38,44);(7,11)(9,12)(13,17)(14,23)(15,28)(16,29)(18,37)(19,36)(20,33)(21,32)(22,27)(24,44)(25,43)(26,40)(30,35)(31,34)(38,42)(39,41);(13,18,20,19)(14,24,26,25)(15,29,31,30)(16,28,35,34)(17,36,33,37)(21,38,39,27)(22,41,42,32)(23,43,40,44)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gh,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, 6366995);