Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417540 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2535 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1152 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,9,16,11,14,10,15)(7,17,19,28,21,12,20,29)(8,22,24,13,26,32,25,31)(18,30,34,33,35,36,23,27);(1,2)(3,4)(5,7)(6,12)(8,23)(9,20)(10,19)(11,21)(13,30)(14,17)(15,29)(16,28)(18,25)(22,33)(24,35)(26,34)(27,32)(31,36);(2,4)(5,8,9,24,11,26,10,25)(6,13,15,22,14,31,16,32)(7,18,19,34,21,35,20,23)(12,27,28,36,17,33,29,30);(12,17)(27,33)(28,29)(30,36)Polycyclic presentation:
2^9|a^2=ef,c^2=f,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,f^b=fi,g^b=gi,h^b=hi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72252923, #72256865, #72256869 |
| maximal normal subgroup of | #72252923, #72256865, #72256869 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 331931);