Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #455962 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2826 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,12,16,11,17,10,30)(7,18,21,13,22,33,23,32)(8,14,31,34,25,20,24,36)(9,26,29,15,28,35,27,19);(1,2,3,4)(5,7,12,21,11,22,10,23)(6,13,16,33,17,32,30,18)(8,15,31,35,25,19,24,26)(9,20,29,36,28,14,27,34);(2,4)(5,8,11,25)(6,14,17,20)(7,19,22,15)(9,18,28,33)(10,31,12,24)(13,29,32,27)(16,36,30,34)(21,35,23,26);(5,9,12,29,11,28,10,27)(6,15,16,35,17,19,30,26)(7,20,21,36,22,14,23,34)(8,13,31,33,25,32,24,18)Polycyclic presentation:
2^9|a^2=ef,b^2=fgh,c^2=i,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,d^c=de,e^c=ei,h^c=hi,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, 380667);