Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| 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 | #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 | |
1092 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,11,16)(7,17)(8,22,26,13)(9,14)(10,15,12,19)(18,34)(20,27)(21,28)(23,36)(24,31)(25,33,29,32)(30,35);(1,2)(3,4)(5,7)(6,9)(8,23)(10,14)(11,21)(12,20)(13,30)(15,28)(16,27)(17,19)(18,25)(22,34)(24,33)(26,31)(29,35)(32,36);(2,4)(5,8,12,29,11,26,10,25)(6,13,19,33,16,22,15,32)(7,18,20,23,21,35,14,31)(9,24,17,34,27,36,28,30);(5,9)(6,14)(7,19)(8,24)(10,28)(11,27)(12,17)(13,31)(15,21)(16,20)(18,33)(22,23)(25,30)(26,36)(29,34)(32,35)Polycyclic presentation:
2^9|a^2=f,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72321897 |
| maximal normal subgroup of | #72321897 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 332212);