Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458126 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #555 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2563 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
945 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,17)(8,22)(9,12)(10,15)(11,16)(13,20)(14,26)(18,31)(19,27)(21,28)(23,30)(24,29)(25,32)(33,35)(34,36);(6,12,16,15)(8,23)(13,28,17,27)(14,30,32,22)(18,34)(24,35)(25,26)(29,31,33,36);(5,7)(6,13,16,17)(8,24)(9,21)(10,19)(11,20)(12,27,15,28)(14,31,32,36)(18,25)(22,33,30,29)(23,35)(26,34);(2,4)(5,8,9,25,11,23,10,26)(6,14,15,30,16,32,12,22)(7,18,19,24,20,34,21,35)(13,29,27,31,17,33,28,36)Polycyclic presentation:
2^9|b^2=g,c^2=g,d^2=e,e^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=di,d^c=de,e^c=ei,f^c=fi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 375331);