Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #427374 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2760 |
| commutator, Frattini, cosocle | C25 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C25 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
1053 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)(7,15)(8,12,20,26)(9,13,21,27)(10,14)(11,18)(16,24,30,22)(17,23,31,28)(19,32)(25,34)(29,36)(33,35);(5,7)(6,11,14,15)(8,16,20,30)(9,17,21,31)(10,18)(12,24)(13,23)(19,33)(22,26)(25,35,32,36)(27,28)(29,34);(5,8)(6,12,14,26)(7,16,18,30)(9,19)(10,20)(11,22)(13,25,27,32)(15,24)(17,29,31,33)(21,34)(23,35)(28,36);(2,4)(5,9,10,21)(6,13)(7,17)(8,19,20,34)(11,23,15,28)(12,25)(14,27)(16,29)(18,31)(22,35,24,36)(26,32)(30,33)Polycyclic presentation:
2^9|a^2=efg,b^2=f,c^2=ef,d^2=efi,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72285222, #72285268, #72285332, … |
| maximal normal subgroup of | #72285222, #72285268, #72285332, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 344911);