Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2598 |
| prime radical | C2×#2017 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6,10,15,11,16,27,17)(7,9,19,28,20,29,14,12)(8,21,24,32,25,13,26,31)(18,23,33,36,34,35,22,30);(2,3)(5,7)(6,12)(8,22)(9,17)(10,14)(11,20)(13,23)(15,29)(16,28)(18,26)(19,27)(21,35)(24,34)(25,33)(30,31)(32,36);(5,8)(6,13)(7,18)(9,23)(10,24)(11,25)(12,30)(14,22)(15,31)(16,21)(17,32)(19,33)(20,34)(26,27)(28,36)(29,35);(1,3,4,2)(5,9,27,28,11,29,10,12)(6,14,17,7,16,19,15,20)(8,23,26,36,25,35,24,30)(13,22,32,18,21,33,31,34)Polycyclic presentation:
2^9|a^2=e,d^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 | #72321148, #72321973 |
| maximal normal subgroup of | #72321148, #72321973 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 335182);