Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500456 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1792 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)(13,14,19,25)(15,26,30,35)(16,22,31,37)(17,24,18,23)(20,29,21,28)(27,40,41,44)(32,39,33,38)(34,43,36,42);(5,6)(7,10)(8,9)(11,12)(13,15,18,29,19,30,17,28)(14,20,24,35,25,21,23,26)(16,27,33,43,31,41,32,42)(22,34,39,44,37,36,38,40);(2,4)(6,9)(7,11)(13,15,17,28,19,30,18,29)(14,21,23,35,25,20,24,26)(16,27,32,42,31,41,33,43)(22,36,38,44,37,34,39,40);(5,7,8,11)(6,10,9,12)(13,16,19,31)(14,22,25,37)(15,27,30,41)(17,32,18,33)(20,34,21,36)(23,38,24,39)(26,40,35,44)(28,42,29,43);(6,9)(10,12)(16,31)(22,37)(27,41)(32,33)(34,36)(38,39)(40,44)(42,43)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=f,d^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421748);