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 | C4×D42 |
| prime radical | #61100 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,7,14,8,15,28,13)(3,16,20,30,21,29,19,9)(4,22,23,5,11,26,24,10)(6,27,18,17,25,12,32,31);(1,3)(2,9)(4,17)(5,25)(6,10)(7,20)(8,21)(11,31)(12,23)(13,29)(14,16)(15,30)(18,22)(19,28)(24,27)(26,32);(1,4)(2,10)(3,17)(5,15)(6,9)(7,23)(8,11)(12,20)(13,26)(14,22)(16,18)(19,27)(21,31)(24,28)(25,30)(29,32);(1,5)(2,11)(3,18)(4,15)(6,19)(7,26)(8,10)(9,27)(12,30)(13,23)(14,24)(16,17)(20,25)(21,32)(22,28)(29,31);(1,6,7,18,8,25,28,32)(2,12,14,31,15,27,13,17)(3,10,20,22,21,5,19,26)(4,9,23,16,11,30,24,29)Polycyclic presentation:
2^9|a^2=h,e^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 | #72285627 |
| maximal normal subgroup of | #72285627 |
| maximal quotient of | #72286208, #72314246, #72342635, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6291082);