Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
963 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)(5,6,9,13,10,14,23,29)(7,15,18,26,19,25,17,11)(8,12,21,27,22,28,32,36)(16,31,34,24,20,30,33,35);(5,7,9,17,10,19,23,18)(6,11,13,25,14,26,29,15)(8,20,21,34,22,16,32,33)(12,24,27,31,28,35,36,30);(5,8)(6,12)(7,16,19,20)(9,21)(10,22)(11,24,26,35)(13,27)(14,28)(15,30,25,31)(17,33,18,34)(23,32)(29,36);(2,4)(7,17,19,18)(11,25,26,15)(16,33,20,34)(24,31,35,30)Polycyclic presentation:
2^9|a^2=e,b^2=ef,c^2=f,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321557 |
| maximal normal subgroup of | #72321557 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 338832);