Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417784 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×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 | |
1037 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)(14,20)(15,25)(17,18)(22,30)(23,24)(26,35)(27,41)(28,29)(31,32)(34,43)(37,38)(39,42);(2,4)(5,7,8,12)(6,9,10,11)(13,14,18,23,19,24,17,20)(15,26,28,36,29,35,25,21)(16,30,32,38,33,37,31,22)(27,34,39,44,42,43,41,40);(1,2,3,4)(6,10)(9,11)(13,15,19,29)(14,21,24,36)(16,27,33,42)(17,28,18,25)(20,26,23,35)(22,34,38,43)(30,40,37,44)(31,39,32,41);(7,12)(9,11)(13,16,19,33)(14,22,24,38)(15,27,29,42)(17,32,18,31)(20,30,23,37)(21,34,36,43)(25,39,28,41)(26,40,35,44)Polycyclic presentation:
2^9|a^2=fg,b^2=ei,c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 341971);