Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #456520 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16283 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1161 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)(3,7)(4,13)(5,10,21,28)(6,11)(8,12)(9,17)(14,22,27,20)(15,30,31,26)(16,25)(18,24)(19,29,32,23);(1,3,7,2,8,11,6,12)(4,13,17,25,18,24,16,9)(5,14,20,10,21,27,22,28)(15,26,19,29,31,30,32,23);(1,4)(2,9)(3,13)(5,19)(6,17)(7,16)(8,18)(10,26)(11,24)(12,25)(14,29)(15,20)(21,32)(22,31)(23,27)(28,30);(1,5)(2,10)(3,14)(4,15)(6,22)(7,20)(8,21)(9,23)(11,27)(12,28)(13,26)(16,32)(17,19)(18,31)(24,30)(25,29)Polycyclic presentation:
2^9|a^2=ef,b^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,e^d=ei53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251786, #72251790, #72251799, … |
| maximal normal subgroup of | #72251786, #72251790, #72251799, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 363716);