Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682440 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2297 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1829 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,18,23)(15,24,19,20)(16,27,30,35)(17,31,34,22)(21,25,38,28)(26,36,32,39)(29,44,33,43)(37,42,40,41);(1,2)(3,4)(6,9)(7,11)(13,15,18,19)(14,20,23,24)(16,28,30,25)(17,32,34,26)(21,27,38,35)(22,39,31,36)(29,41,33,42)(37,44,40,43);(2,4)(13,16,15,25,18,30,19,28)(14,21,20,35,23,38,24,27)(17,33,26,42,34,29,32,41)(22,40,36,43,31,37,39,44);(6,9)(10,12)(13,17)(14,22)(15,26)(16,29)(18,34)(19,32)(20,36)(21,37)(23,31)(24,39)(25,41)(27,43)(28,42)(30,33)(35,44)(38,40);(2,4)(5,7,8,11)(6,10,9,12)(16,30)(17,34)(21,38)(22,31)(25,28)(26,32)(27,35)(36,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=gi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6592683);