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 | #45091 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2174 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)(13,14,19,24)(15,26,28,21)(16,29,32,40)(17,34,36,42)(18,25,20,23)(22,31,41,33)(27,38,43,44)(30,37,35,39);(1,2)(3,4)(6,10)(11,12)(13,15,19,28)(14,21,24,26)(16,30,32,35)(17,20,36,18)(22,38,41,44)(23,34,25,42)(27,33,43,31)(29,39,40,37);(5,7,9,8)(6,11,10,12)(13,16,18,31,19,32,20,33)(14,22,23,40,24,41,25,29)(15,27,36,30,28,43,17,35)(21,37,42,38,26,39,34,44);(2,4)(5,8,9,7)(6,12,10,11)(13,17,20,28,19,36,18,15)(14,21,25,34,24,26,23,42)(16,30,33,27,32,35,31,43)(22,39,29,38,41,37,40,44);(6,10)(11,12)(13,18,19,20)(14,23,24,25)(15,17,28,36)(16,31,32,33)(21,34,26,42)(22,40,41,29)(27,35,43,30)(37,44,39,38)Polycyclic presentation:
2^9|a^2=i,b^2=i,c^2=fg,d^2=fgi,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, 7469470);