Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394465 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#124 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | C2×#394 |
| prime radical | C22×#575 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3536 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)(7,15,17,11)(8,19,21,12)(9,14)(10,13)(16,33)(18,31,23,27)(20,36)(22,32,24,29)(25,30)(26,34)(28,35);(5,7)(6,11)(8,20)(9,23)(10,17)(12,28)(13,15)(14,31)(16,22)(18,25)(19,36)(21,35)(24,34)(26,29)(27,30)(32,33);(5,8)(6,12)(7,16)(9,24)(10,21)(11,26)(13,19)(14,32)(15,33)(17,34)(18,35)(20,23)(22,25)(27,36)(28,31)(29,30);(2,4)Polycyclic presentation:
2^9|a^2=ef,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,g^b=gi,e^c=ei,g^c=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72257114, #72257140, #72257150 |
| maximal normal subgroup of | #72257114, #72257140, #72257150 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302574);