Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394127 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2410 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3940 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)(5,6,11,16)(7,17,22,29)(8,23,25,32)(9,15,10,14)(12,21,30,20)(13,26,31,24)(18,34,36,27)(19,28,35,33);(2,4)(5,7)(6,12)(8,19)(9,21)(10,20)(11,22)(13,28)(14,29)(15,17)(16,30)(18,24)(23,27)(25,35)(26,36)(31,33)(32,34);(5,8)(6,13)(7,18)(9,26)(10,24)(11,25)(12,27)(14,32)(15,23)(16,31)(17,33)(19,21)(20,35)(22,36)(28,29)(30,34);(5,8)(6,13)(7,19)(9,24)(10,26)(11,25)(12,28)(14,23)(15,32)(16,31)(17,34)(18,21)(20,36)(22,35)(27,29)(30,33)Polycyclic presentation:
2^9|a^2=i,e^2=i,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,e^b=ei,g^b=gi,e^c=ei,f^c=fi,g^c=gi,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72255736, #72285083, #72285466, … |
| maximal normal subgroup of | #72255736, #72285083, #72285466, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301325);