Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417603 |
| 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 | |
1113 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 44 points:
(2,4)(5,6,9,11)(7,12,8,10)(13,14,19,25)(15,26,17,34)(16,29,33,22)(18,36,20,24)(21,35,23,28)(27,43,31,39)(30,38,44,40)(32,41,37,42);(1,2,3,4)(13,15,18,28,19,17,20,35)(14,21,24,26,25,23,36,34)(16,30,32,27,33,44,37,31)(22,39,41,40,29,43,42,38);(5,7,9,8)(6,10,11,12)(13,16,19,33)(14,22,25,29)(15,27,17,31)(18,32,20,37)(21,38,23,40)(24,41,36,42)(26,39,34,43)(28,44,35,30);(2,4)(5,8,9,7)(6,10,11,12)(13,17)(14,23)(15,19)(16,31)(18,28)(20,35)(21,25)(22,40)(24,26)(27,33)(29,38)(30,37)(32,44)(34,36)(39,41)(42,43)Polycyclic presentation:
2^9|a^2=f,b^2=e,c^2=f,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 338280);