Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418860 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15266 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #60367 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1972 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,10,7,12,6,11)(3,13,15,25,17,24,16,8)(4,18,20,27,22,9,21,28)(14,29,19,23,31,26,30,32);(1,3)(2,8)(4,19)(5,15)(6,16)(7,17)(9,26)(10,13)(11,24)(12,25)(14,21)(18,29)(20,31)(22,30)(23,27)(28,32);(1,4)(2,9)(3,14)(5,21)(6,20)(7,22)(8,23)(10,27)(11,28)(12,18)(13,29)(15,30)(16,19)(17,31)(24,26)(25,32);(2,10)(3,15)(5,6)(9,27)(11,12)(13,24)(14,30)(16,17)(18,28)(19,31)(20,21)(26,29)Polycyclic presentation:
2^9|a^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=gi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72216013, #72216154, #72216217, … |
| maximal normal subgroup of | #72216013, #72216154, #72216217, … |
| maximal quotient of | C2×#418860, #72252176, #72255762, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326056);