Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395191 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2431 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #14905 |
| prime radical | C2×#525 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1151 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)(3,15)(4,10)(5,22)(6,14)(7,13)(8,12)(9,18)(11,23)(16,31)(17,29)(19,32)(20,30)(21,28)(24,26)(25,27);(1,3,8,25,7,19,6,18)(2,9,14,32,13,27,12,15)(4,16,20,5,17,23,21,24)(10,22,28,31,29,26,30,11);(1,4,8,20,7,17,6,21)(2,10,14,28,13,29,12,30)(3,16,25,5,19,23,18,24)(9,26,32,11,27,22,15,31);(1,5)(2,11,13,31)(3,17)(4,19)(6,16)(7,24)(8,23)(9,10,27,29)(12,26,14,22)(15,30,32,28)(18,20)(21,25)Polycyclic presentation:
2^9|b^2=ef,c^2=ef,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72213514, #72215912, #72216130, … |
| maximal normal subgroup of | #72213514, #72215912, #72216130, … |
| maximal quotient of | #72250605, C2×#404881, #72256021, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 312077);