Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6374204 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#2308 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62852 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
4407 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,13,17,8)(4,18,20,27)(5,10)(6,12)(7,11)(9,19,26,21)(14,28)(15,24,16,23)(22,29)(25,31)(30,32);(1,3)(2,8)(4,14)(5,16)(6,15)(7,17)(9,25)(10,24)(11,13)(12,23)(18,32)(19,31)(20,30)(21,29)(22,26)(27,28);(1,4)(2,9)(3,14)(5,19)(6,21)(7,20)(8,22)(10,18)(11,26)(12,27)(13,25)(15,29)(16,31)(17,30)(23,32)(24,28);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,22)(10,23)(11,13)(12,24)(18,32)(19,29)(20,30)(21,31)(25,26)(27,28);(1,5,7,6)(2,10,11,12)(3,15,17,16)(4,19,20,21)(8,23,13,24)(9,18,26,27)(14,29,30,31)(22,32,25,28)Polycyclic presentation:
2^9|a^2=h,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,f^a=fi,c^b=cf,e^b=ei,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72240646, #72249649, #72251416, … |
| maximal normal subgroup of | #72240646, #72249649, #72251416, … |
| maximal quotient of | #72255780, #72256081, #72262221 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6281402);