Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394113 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2403 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3585 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3932 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)(5,6,10,16)(7,17,21,13)(8,22,24,30)(9,12,11,15)(14,25,29,23)(18,28,20,26)(19,33,34,36)(27,35,31,32);(6,12)(7,18)(9,11)(14,22)(15,16)(19,32)(20,21)(23,25)(26,28)(29,30)(33,36)(34,35);(2,4)(5,7)(6,13)(8,19)(9,18)(10,21)(11,20)(12,26)(14,27)(15,28)(16,17)(22,36)(23,35)(24,34)(25,32)(29,31)(30,33);(5,8)(6,14)(7,19)(9,25)(10,24)(11,23)(12,22)(13,27)(15,30)(16,29)(17,31)(18,32)(20,35)(21,34)(26,36)(28,33)Polycyclic presentation:
2^9|a^2=i,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei,h^d=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72338205 |
| maximal normal subgroup of | #72338205 |
| maximal quotient of | #72338256 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301321);