Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417784 |
| 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 | |
1219 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)(5,6,8,10)(7,11,12,9)(14,19)(16,29)(17,20)(21,33)(23,34)(24,36)(25,40)(26,27)(30,37)(31,32)(39,44)(42,43);(1,2,3,4)(13,14,20,34,18,23,17,19)(15,24,27,21,28,33,26,36)(16,30,32,38,31,37,29,22)(25,41,40,44,43,35,42,39);(5,7,8,12)(6,9,10,11)(13,15)(14,21)(16,25)(17,27)(18,28)(19,33)(20,26)(22,35)(23,36)(24,34)(29,40)(30,44)(31,43)(32,42)(37,39)(38,41);(2,4)(13,16,18,31)(14,22,23,38)(15,25,28,43)(17,32,20,29)(19,30,34,37)(21,35,36,41)(24,39,33,44)(26,40,27,42)Polycyclic presentation:
2^9|a^2=f,b^2=ei,c^2=f,d^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=dei,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332120);