Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #346446 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2846 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | #62818 | C2 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2530 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)(3,11,16,31)(4,17,19,6)(5,18,23,21)(8,24,26,13)(9,27,29,22)(12,32,25,20)(14,30,15,28);(1,3)(2,8)(4,12)(5,20)(6,15)(7,16)(9,28)(10,26)(11,18)(13,22)(14,17)(19,25)(21,31)(23,32)(24,27)(29,30);(1,4,17,10)(2,7,19,6)(3,12,14,26)(5,21,27,9)(8,16,25,15)(11,13,30,32)(18,22,29,23)(20,31,24,28);(5,22)(9,18)(11,28)(13,20)(21,29)(23,27)(24,32)(30,31)Polycyclic presentation:
2^9|a^2=i,c^2=hi,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh59 conjugacy classes — view the full character table.
| maximal subgroup of | #72213843, #72232843, #72232857, … |
| maximal normal subgroup of | #72213843, #72232843, #72232857, … |
| maximal quotient of | C2×#346471, #72261571 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 253667);