Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454664 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2622 |
| 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 | |
993 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,15,10,14,26,13)(7,16,25,29,19,28,18,11)(8,20,23,31,24,12,22,32)(17,30,33,27,34,35,21,36);(1,2,3,4)(5,7,10,19)(6,11,14,29)(8,21,24,33)(9,25,26,18)(12,30,20,35)(13,28,15,16)(17,22,34,23)(27,31,36,32);(2,4)(5,8,10,24)(6,12,14,20)(7,17,19,34)(9,22,26,23)(11,27,29,36)(13,32,15,31)(16,30,28,35)(18,33,25,21);(6,13)(8,22)(9,26)(11,28)(12,20)(14,15)(16,29)(17,33)(18,25)(21,34)(23,24)(30,35)Polycyclic presentation:
2^9|a^2=ef,b^2=gh,c^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,g^c=gi,h^c=hi,e^d=ei,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, 378980);