Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454661 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16255 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3021 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1807 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,6,12,5,11)(3,13,15,8,16,24,20,23)(4,17,21,9,19,25,18,26)(14,28,29,32,30,22,31,27);(1,3)(2,8)(4,14)(5,20)(6,16)(7,15)(9,22)(10,24)(11,13)(12,23)(17,32)(18,31)(19,30)(21,29)(25,27)(26,28);(4,18)(5,7)(9,25)(10,11)(13,24)(14,29)(15,20)(17,26)(19,21)(22,32)(27,28)(30,31);(1,4)(2,9)(3,14)(5,21)(6,19)(7,18)(8,22)(10,17)(11,25)(12,26)(13,27)(15,31)(16,30)(20,29)(23,28)(24,32)Polycyclic presentation:
2^9|a^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,h^c=hi,e^d=ei,g^d=gi,h^d=hi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237064, #72237072, #72237092, … |
| maximal normal subgroup of | #72237064, #72237072, #72237092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361857);