Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458076 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16309 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
953 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8)(3,12)(5,6)(9,13)(10,26)(11,21)(14,29)(18,20)(22,25)(23,28)(24,27)(30,32);(1,2,5,8,7,11,6,21)(3,13,16,9,12,22,15,25)(4,10,20,23,19,28,18,26)(14,30,17,24,29,27,31,32);(1,3,5,16,7,12,6,15)(2,9,8,22,11,25,21,13)(4,17,20,29,19,31,18,14)(10,27,23,32,28,30,26,24);(1,4)(2,10)(3,14)(5,20)(6,18)(7,19)(8,23)(9,24)(11,28)(12,29)(13,30)(15,31)(16,17)(21,26)(22,27)(25,32)Polycyclic presentation:
2^9|b^2=ef,c^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,c^b=cg,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72251783, #72251790, #72251797, … |
| maximal normal subgroup of | #72251783, #72251790, #72251797, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365272);