Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6628851 |
| 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 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4014 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)(5,6,8,10)(7,11,12,9)(13,14,18,20)(15,24,27,36)(16,22,29,34)(17,23,19,31)(21,32,33,26)(25,39,41,44)(28,37,30,38)(35,40,43,42);(5,7,8,12)(6,9,10,11)(14,20)(21,33)(22,34)(23,31)(24,36)(35,43)(37,38)(39,44);(13,15,18,27)(14,21,20,33)(16,25,29,41)(17,26,19,32)(22,35,34,43)(23,36,31,24)(28,40,30,42)(37,39,38,44);(2,4)(13,16)(14,22)(15,25)(17,30)(18,29)(19,28)(20,34)(21,35)(23,38)(24,39)(26,42)(27,41)(31,37)(32,40)(33,43)(36,44);(13,17,18,19)(14,23,20,31)(15,26,27,32)(16,28,29,30)(21,36,33,24)(22,37,34,38)(25,40,41,42)(35,39,43,44)Polycyclic presentation:
2^9|a^2=f,b^2=fi,c^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,g^c=gi,e^d=ei98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6536292);