Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #403988 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14885 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2620 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,13)(4,17)(5,11)(6,9)(7,12)(8,15)(10,20)(14,28)(16,24)(18,25)(19,26)(21,23)(22,32)(27,30)(29,31);(1,3)(2,8)(4,14)(5,21)(6,15)(7,16)(9,24)(10,22)(11,13)(12,23)(17,31)(18,32)(19,29)(20,30)(25,28)(26,27);(2,9)(5,6)(8,13)(10,25)(11,12)(15,21)(17,26)(18,19)(22,27)(23,24)(28,31)(29,32);(1,4)(2,10,12,17)(3,14)(5,18)(6,19)(7,20)(8,22,23,31)(9,25,11,26)(13,27,24,28)(15,29)(16,30)(21,32)Polycyclic presentation:
2^9|d^2=g,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,e^c=ei,f^c=fi,h^d=hi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72233772, #72233773, #72233778, … |
| maximal normal subgroup of | #72233772, #72233773, #72233778, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 311184);