Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417704 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1714 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)(5,6,11,14,9,16,10,15)(7,12,20,28,18,29,19,27)(8,21,25,32,23,31,24,13)(17,33,22,36,34,30,35,26);(5,7,9,18)(6,12,16,29)(8,22,23,35)(10,20,11,19)(13,30,32,33)(14,27,15,28)(17,24,34,25)(21,36,31,26);(2,4)(5,8,9,23)(6,13)(7,17,18,34)(10,24,11,25)(12,26)(14,21)(15,31)(16,32)(19,35,20,22)(27,30)(28,33)(29,36);(6,14,16,15)(7,18)(12,27,29,28)(13,21,32,31)(17,34)(19,20)(22,35)(26,30,36,33)Polycyclic presentation:
2^9|a^2=ef,b^2=i,c^2=e,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250742, #72250763, #72250789, … |
| maximal normal subgroup of | #72250742, #72250763, #72250789, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327696);