Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61106 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×D42 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| socle | C2 | C2×#2358 |
| prime radical | #60387 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,12,6,10,21,11)(3,9,15,23,16,24,14,7)(4,17,18,26,20,27,19,8)(13,28,30,32,31,22,29,25);(1,3,5,15,6,16,21,14)(2,7,12,9,10,23,11,24)(4,13,18,30,20,31,19,29)(8,22,17,25,26,28,27,32);(1,4)(2,8)(3,13)(5,18)(6,20)(7,22)(9,25)(10,26)(11,27)(12,17)(14,29)(15,30)(16,31)(19,21)(23,28)(24,32);(1,3)(2,9)(4,13)(5,15)(6,16)(7,11)(8,25)(10,24)(12,23)(14,21)(17,28)(18,30)(19,29)(20,31)(22,27)(26,32);(2,10)(7,23)(8,26)(9,24)(11,12)(17,27)(22,28)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,g^a=gi,d^b=df,f^b=fi,g^c=gi,f^d=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72213889, #72214082, #72216141, … |
| maximal normal subgroup of | #72213889, #72214082, #72216141, … |
| maximal quotient of | #72286392, #72342639, #72342650 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6288680);