Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455962 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C24 | 2−1+4 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,13,10,14,20,24)(7,11,17,15,18,27,25,32)(8,12,21,29,22,31,23,30)(16,26,19,28,33,36,34,35);(1,2,3,4)(5,6)(7,15)(8,12)(9,24)(10,14)(11,17)(13,20)(16,28)(18,32)(19,26)(21,30)(22,31)(23,29)(25,27)(33,35)(34,36);(2,4)(5,7,10,18)(6,11,14,27)(8,19,22,34)(9,25,20,17)(12,28,31,35)(13,32,24,15)(16,23,33,21)(26,30,36,29);(5,8,20,23,10,22,9,21)(6,12,13,29,14,31,24,30)(7,16,25,34,18,33,17,19)(11,26,15,28,27,36,32,35)Polycyclic presentation:
2^9|a^2=e,b^2=egh,c^2=i,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,g^c=gi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 380152);