Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #418872 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #15266 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1976 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,7,10,6,11,5,20)(3,12,14,8,15,27,21,24)(4,16,22,28,19,9,18,26)(13,25,17,23,30,29,31,32);(1,3)(2,8)(4,17)(5,21)(6,15)(7,14)(9,25)(10,27)(11,24)(12,20)(13,18)(16,29)(19,31)(22,30)(23,26)(28,32);(5,7)(10,20)(12,27)(14,21)(17,31)(18,22)(25,29)(26,28);(1,4)(2,9)(3,13)(5,22)(6,19)(7,18)(8,23)(10,28)(11,16)(12,29)(14,31)(15,30)(17,21)(20,26)(24,32)(25,27)Polycyclic presentation:
2^9|a^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,e^d=ei,g^d=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72286635, #72287064, #72287267, … |
| maximal normal subgroup of | #72286635, #72287064, #72287267, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 326068);