Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417692 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | H(5,𝔽2) |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1175 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(1,2)(3,4)(6,9)(10,11)(13,14,18,20,19,24,17,23)(15,25,28,37,29,34,27,21)(16,30,32,38,33,35,31,22)(26,36,41,43,42,44,40,39);(5,6,8,9)(7,11,12,10)(14,20)(16,31)(17,18)(21,34)(22,38)(23,24)(25,37)(26,40)(27,28)(32,33)(39,43)(41,42);(13,15,19,29)(14,21,24,37)(16,26,33,42)(17,28,18,27)(20,34,23,25)(22,36,38,44)(30,39,35,43)(31,41,32,40);(2,4)(5,7,8,12)(6,10,9,11)(13,16)(14,22)(15,26)(17,32)(18,31)(19,33)(20,35)(21,36)(23,30)(24,38)(25,39)(27,41)(28,40)(29,42)(34,43)(37,44)Polycyclic presentation:
2^9|a^2=f,b^2=ef,c^2=i,d^2=ef,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,f^b=fi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 337675);