Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667932 |
| 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×#525 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2101 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(5,6)(7,10)(8,9)(11,12)(13,14,17,22)(15,24,26,19)(16,28,29,21)(18,20,31,23)(25,33,38,37)(27,34,40,32)(30,36,41,35)(39,42,44,43);(1,2)(3,4)(6,9)(7,11)(13,15,17,26)(14,19,22,24)(16,25,29,38)(18,27,31,40)(20,32,23,34)(21,33,28,37)(30,39,41,44)(35,42,36,43);(2,4)(5,7,8,11)(6,10,9,12)(14,20)(18,31)(19,32)(21,35)(22,23)(24,34)(27,40)(28,36)(30,41)(33,42)(37,43)(39,44);(6,9)(10,12)(13,16,17,29)(14,21,22,28)(15,25,26,38)(18,30,31,41)(19,33,24,37)(20,35,23,36)(27,39,40,44)(32,42,34,43);(2,4)(16,29)(21,28)(25,38)(30,41)(33,37)(35,36)(39,44)(42,43)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6576454);