Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #381470 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14299 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | #23400 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1415 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)(3,12)(4,16,20,17)(5,21)(6,10)(7,11)(8,14)(9,19,27,18)(13,26,30,23)(15,24)(22,25)(28,29,32,31);(1,3)(2,8)(4,13)(5,22)(6,14)(7,15)(9,26)(10,24)(11,25)(12,21)(16,32)(17,28)(18,31)(19,29)(20,30)(23,27);(1,4)(2,9)(3,13)(5,19)(6,18)(7,20)(8,23)(10,17)(11,27)(12,28)(14,31)(15,30)(16,21)(22,29)(24,32)(25,26);(1,2)(3,8)(4,17,20,16)(5,10)(6,21)(7,11)(9,19,27,18)(12,14)(13,28,30,32)(15,25)(22,24)(23,31,26,29)Polycyclic presentation:
2^9|a^2=h,d^2=h,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,g^a=gi,c^b=cg,e^b=ei,d^c=de,e^c=ei,f^c=fi,h^c=hi,f^d=fi,g^d=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72216013, #72216077, #72216142, … |
| maximal normal subgroup of | #72216013, #72216077, #72216142, … |
| maximal quotient of | C2×#381470, #72252099 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 288666);