Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #420522 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15278 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,13,7,11,6,12)(3,14,17,25,19,8,18,26)(4,10,20,9,21,29,22,28)(15,30,16,24,31,27,32,23);(1,3)(2,8)(4,16)(5,18)(6,17)(7,19)(9,27)(10,23)(11,14)(12,26)(13,25)(15,20)(21,32)(22,31)(24,29)(28,30);(1,4)(2,9,11,28)(3,15)(5,20)(6,22)(7,21)(8,23,14,24)(10,13,29,12)(16,17)(18,32)(19,31)(25,27,26,30);(1,4)(2,10,11,29)(3,16)(5,22)(6,20)(7,21)(8,24,14,23)(9,13,28,12)(15,17)(18,31)(19,32)(25,27,26,30)Polycyclic presentation:
2^9|a^2=ef,c^2=e,d^2=e,f^2=i,g^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,d^b=de,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,f^d=fi,h^d=hi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72316368, C2×#420522, #72319466 |
| maximal normal subgroup of | #72316368, C2×#420522, #72319466 |
| maximal quotient of | C2×#420522, #72316828, #72316834, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 327718);