Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417570 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2580 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
914 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(5,6,10,15,11,16,19,17)(7,18,21,33,22,32,30,12)(8,23,25,31,26,13,27,35)(9,28,29,36,24,14,20,34);(1,2)(3,4)(5,7,19,21,11,22,10,30)(6,12,17,18,16,33,15,32)(8,24,27,20,26,9,25,29)(13,34,31,28,23,36,35,14);(2,4)(5,8)(6,13)(7,20)(9,21)(10,25)(11,26)(12,14)(15,35)(16,23)(17,31)(18,34)(19,27)(22,29)(24,30)(28,33)(32,36);(5,9,19,29,11,24,10,20)(6,14,17,34,16,28,15,36)(7,8,21,27,22,26,30,25)(12,31,18,23,33,35,32,13)Polycyclic presentation:
2^9|a^2=e,b^2=ef,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 | #72322900 |
| maximal normal subgroup of | #72322900 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336872);