Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #419843 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2688 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×AGL(1,ℤ/8) | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1355 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(6,12)(7,17)(9,10)(14,27)(15,16)(18,26)(19,33)(20,21)(22,30)(23,24)(28,36)(31,34);(1,2)(3,4)(5,6)(7,18)(8,22)(9,15)(10,12)(11,16)(13,20)(14,23)(17,29)(19,28)(21,26)(24,30)(25,27)(31,32)(33,35)(34,36);(2,4)(5,7)(6,13,16,29)(8,19)(9,17)(10,20)(11,21)(12,26,15,18)(14,28,30,36)(22,32,27,35)(23,33)(24,31)(25,34);(5,8,11,25)(6,14,16,30)(7,19,21,34)(9,24,10,23)(12,27,15,22)(13,28,29,36)(17,31,20,33)(18,32,26,35)Polycyclic presentation:
2^9|c^2=ef,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=de,e^b=ei,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72306351, #72306456, #72306464 |
| maximal normal subgroup of | #72306351, #72306456, #72306464 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 330383);