Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #393925 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1873 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)(5,6,8,10)(7,11,12,9)(13,14,18,20,19,24,17,23)(15,25,28,35,29,34,26,21)(16,30,32,22,33,37,31,38)(27,41,40,39,43,36,42,44);(2,4)(5,7,8,12)(6,9,10,11)(14,20)(15,26)(17,18)(21,35)(22,30)(23,24)(27,40)(28,29)(31,32)(36,41)(37,38)(42,43);(2,4)(13,15,19,29)(14,21,24,35)(16,27,33,43)(17,28,18,26)(20,34,23,25)(22,36,38,41)(30,44,37,39)(31,40,32,42);(13,16)(14,22)(15,27)(17,32)(18,31)(19,33)(20,30)(21,36)(23,37)(24,38)(25,39)(26,40)(28,42)(29,43)(34,44)(35,41)Polycyclic presentation:
2^9|a^2=ei,b^2=ef,c^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,h^d=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 303757);