Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454002 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3005 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1849 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)(7,17)(8,22)(9,15)(10,14)(11,16)(12,19)(13,24)(18,33)(20,29)(21,28)(23,36)(25,31)(26,32)(27,35)(30,34);(5,7,11,21)(6,12,16,29)(8,23,26,34)(9,20,10,19)(13,30,31,36)(14,28,15,17)(18,24,35,25)(22,27,32,33);(2,4)(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,27)(14,22)(15,32)(16,31)(17,30)(19,34)(20,23)(21,35)(28,36)(29,33);(6,14)(8,24)(9,10)(12,28)(13,31)(15,16)(17,29)(18,34)(19,20)(23,35)(25,26)(30,36)Polycyclic presentation:
2^9|a^2=ef,b^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,h^a=hi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi,h^d=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72285010, #72285507, #72285571, … |
| maximal normal subgroup of | #72285010, #72285507, #72285571, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361892);