Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #394022 |
| 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 | |
3990 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)(5,6,11,16)(7,17,21,26)(8,13,23,29)(9,15,10,14)(12,20,27,18)(19,33,35,31)(22,30,24,28)(25,32,36,34);(5,7)(6,12)(8,19)(9,20)(10,18)(11,21)(13,25)(14,26)(15,17)(16,27)(22,32)(23,35)(24,34)(28,31)(29,36)(30,33);(2,4)(7,18)(9,10)(12,17)(14,15)(19,32)(20,21)(22,24)(25,31)(26,27)(28,30)(33,36)(34,35);(5,8)(6,13)(7,19)(9,24)(10,22)(11,23)(12,25)(14,30)(15,28)(16,29)(17,31)(18,32)(20,34)(21,35)(26,33)(27,36)Polycyclic presentation:
2^9|a^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,e^b=ei,g^b=gi,h^b=hi,e^c=ei,g^c=gi,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239341, #72302579, #72302608, … |
| maximal normal subgroup of | #72239341, #72302579, #72302608, … |
| maximal quotient of | #72280719, #72316697, #72316781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 301220);