Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6500535 |
| 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 | |
1803 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)(13,14,20,26)(15,22,29,37)(16,21,28,38)(17,23,33,42)(18,25,19,24)(27,39,31,41)(30,36,32,40)(34,44,35,43);(5,6)(7,10)(8,9)(11,12)(13,15,19,28,20,29,18,16)(14,21,25,37,26,38,24,22)(17,31,35,30,33,27,34,32)(23,40,44,39,42,36,43,41);(2,4)(6,9)(7,11)(13,16,18,29,20,28,19,15)(14,22,24,38,26,37,25,21)(17,32,34,27,33,30,35,31)(23,41,43,36,42,39,44,40);(5,7,8,11)(6,10,9,12)(13,17)(14,23)(15,27)(16,30)(18,34)(19,35)(20,33)(21,36)(22,39)(24,43)(25,44)(26,42)(28,32)(29,31)(37,41)(38,40);(6,9)(10,12)(17,33)(23,42)(27,31)(30,32)(34,35)(36,40)(39,41)(43,44)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,d^b=di,e^b=eh,d^c=dhi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6421732);