Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #401009 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1708 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)(7,12,18,27)(8,20,21,35)(9,24)(10,15)(11,14)(13,23,29,22)(16,28,19,25)(17,31)(26,32)(30,36)(33,34);(1,2)(3,4)(5,7,11,18)(6,12)(8,17,21,33)(9,16,10,19)(13,26)(14,27)(15,25)(20,34)(22,30,23,32)(24,28)(29,36)(31,35);(2,4)(7,16)(9,10)(12,25)(15,24)(17,30)(18,19)(20,35)(22,23)(26,34)(27,28)(31,36)(32,33);(5,8)(6,13,14,29)(7,17)(9,22)(10,23)(11,21)(12,26,27,36)(15,20,24,35)(16,30)(18,33)(19,32)(25,34,28,31)Polycyclic presentation:
2^9|a^2=f,b^2=efi,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,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 | #72255617, #72256282, #72256289 |
| maximal normal subgroup of | #72255617, #72256282, #72256289 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 308206);