Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395174 |
| 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 | #2482 |
| prime radical | #3449 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ 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)(7,11,17,25)(8,19,20,12)(9,13)(10,14)(15,22)(16,27)(18,24,32,26)(21,28,33,29)(23,30)(31,34)(35,36);(1,2)(3,4)(5,7,9,18,10,17,22,32)(6,11,15,26,14,25,13,24)(8,16,21,31,20,30,33,36)(12,27,28,35,19,23,29,34);(2,4)(5,8)(6,12)(7,16)(9,21)(10,20)(11,23)(13,29)(14,19)(15,28)(17,30)(18,31)(22,33)(24,35)(25,27)(26,34)(32,36);(6,13,14,15)(8,20)(11,24,25,26)(12,28,19,29)(16,30)(21,33)(23,34,27,35)(31,36)Polycyclic presentation:
2^9|a^2=ef,b^2=h,d^2=f,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,e^b=ei,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72265222, #72307328, #72307350, … |
| maximal normal subgroup of | #72265222, #72307328, #72307350, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 315052);