Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514040 |
| 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 | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1812 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)(7,12)(8,10)(9,11)(13,14,18,23)(15,25,28,20)(16,30,33,40)(17,24,19,21)(22,34,39,32)(26,36,29,37)(27,42,43,38)(31,41,44,35);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,15,18,28)(14,20,23,25)(16,31,33,44)(17,29,19,26)(21,37,24,36)(22,38,39,42)(27,34,43,32)(30,35,40,41);(5,8,9,7)(6,10,11,12)(14,21)(15,26)(17,19)(22,30)(23,24)(27,31)(28,29)(32,34)(36,37)(38,42)(39,40)(43,44);(2,4)(13,16,18,33)(14,22,23,39)(15,27,28,43)(17,32,19,34)(20,35,25,41)(21,30,24,40)(26,31,29,44)(36,38,37,42);(6,11)(10,12)(13,17,18,19)(14,21,23,24)(15,26,28,29)(16,32,33,34)(20,36,25,37)(22,30,39,40)(27,31,43,44)(35,38,41,42)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fi,d^2=i,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=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,h^c=hi,g^d=gi,h^d=hi71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7422887);