Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7519008 |
| 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×#547 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1823 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)(13,14,20,26)(15,21,17,35)(16,30,34,40)(18,24,19,25)(22,33,41,32)(23,29,38,28)(27,42,36,39)(31,44,43,37);(1,2)(3,4)(6,9)(10,12)(13,15,20,17)(14,21,26,35)(16,31,34,43)(18,28,19,29)(22,39,41,42)(23,25,38,24)(27,32,36,33)(30,44,40,37);(5,7,8,11)(6,10,9,12)(13,16,18,32,20,34,19,33)(14,22,25,30,26,41,24,40)(15,27,29,43,17,36,28,31)(21,37,23,42,35,44,38,39);(2,4)(13,17,19,29,20,15,18,28)(14,23,24,21,26,38,25,35)(16,27,33,31,34,36,32,43)(22,39,40,44,41,42,30,37);(6,9)(10,12)(13,18,20,19)(14,24,26,25)(15,28,17,29)(16,32,34,33)(21,23,35,38)(22,40,41,30)(27,31,36,43)(37,42,44,39)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=gi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7468382);