Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420055 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2476 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #15391 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
947 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,12,8,13,6,26)(3,14,16,27,17,32,15,9)(4,18,20,28,21,30,19,10)(5,22,24,31,25,11,23,29);(1,3,6,15,8,17,7,16)(2,9,12,14,13,27,26,32)(4,5,19,23,21,25,20,24)(10,11,18,29,28,22,30,31);(1,4)(2,10)(3,5)(6,20)(7,19)(8,21)(9,22)(11,27)(12,30)(13,28)(14,29)(15,24)(16,23)(17,25)(18,26)(31,32);(1,5)(2,11,13,22)(3,4)(6,24)(7,23)(8,25)(9,10,27,28)(12,31,26,29)(14,30,32,18)(15,20)(16,19)(17,21)Polycyclic presentation:
2^9|a^2=f,b^2=e,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,f^d=fi,g^d=gi,h^d=hi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72319831, #72319976, #72319988, … |
| maximal normal subgroup of | #72319831, #72319976, #72319988, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 336068);