Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(13,14,19,24)(15,25,29,35)(16,30,33,37)(17,23,18,22)(20,28,36,27)(21,32,38,31)(26,40,43,34)(39,42,44,41);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,20,24,36)(16,26,33,43)(17,28,18,27)(21,34,38,40)(22,35,23,25)(30,44,37,39)(31,42,32,41);(6,9)(10,12)(13,16)(14,21)(15,26)(17,31)(18,32)(19,33)(20,34)(22,30)(23,37)(24,38)(25,39)(27,41)(28,42)(29,43)(35,44)(36,40);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,33,32)(20,25,36,35)(21,30,38,37)(26,41,43,42)(34,39,40,44);(7,11)(10,12)(13,17,19,18)(14,23,24,22)(15,27,29,28)(16,31,33,32)(20,35,36,25)(21,37,38,30)(26,41,43,42)(34,44,40,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fi,g^a=gi,c^b=ch,d^b=di,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6440534);