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 | |
2176 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)(15,23)(16,30)(17,35)(18,24)(19,26)(20,25)(21,28)(22,33)(27,38)(29,37)(31,36)(32,41)(34,40)(39,43)(42,44);(1,2)(3,4)(6,10)(11,12)(13,15,18,28)(14,21,24,23)(16,31,32,43)(17,20,29,19)(22,38,40,44)(25,35,26,37)(27,33,42,34)(30,39,41,36);(5,7,9,8)(6,11,10,12)(13,16,19,33,18,32,20,34)(14,22,25,30,24,40,26,41)(15,27,29,43,28,42,17,31)(21,36,37,44,23,39,35,38);(2,4)(5,8,9,7)(6,12,10,11)(13,17,20,28,18,29,19,15)(14,23,26,37,24,21,25,35)(16,31,34,42,32,43,33,27)(22,39,41,44,40,36,30,38);(6,10)(11,12)(14,24)(16,32)(21,23)(25,26)(27,42)(31,43)(33,34)(35,37)Polycyclic presentation:
2^9|b^2=i,c^2=fgi,d^2=fg,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^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7469359);