Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7518944 |
| 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 ◁ #2398 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1813 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)(7,11)(8,12)(9,10)(13,14,19,25)(15,27,31,21)(16,33,35,39)(17,36,38,43)(18,26,20,24)(22,29,42,34)(23,37,44,28)(30,40,32,41);(2,4)(6,10)(11,12)(13,15,19,31)(14,21,25,27)(16,17,35,38)(18,30,20,32)(22,23,42,44)(24,40,26,41)(28,34,37,29)(33,43,39,36);(5,7,9,8)(6,11,10,12)(13,16,20,29,19,35,18,34)(14,22,26,39,25,42,24,33)(15,28,30,17,31,37,32,38)(21,36,40,23,27,43,41,44);(1,2,3,4)(13,17,18,28,19,38,20,37)(14,23,24,36,25,44,26,43)(15,29,32,16,31,34,30,35)(21,39,41,22,27,33,40,42);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,24,25,26)(15,30,31,32)(16,34,35,29)(17,37,38,28)(21,40,27,41)(22,33,42,39)(23,43,44,36)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=hi,e^2=f,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,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7521602);