Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #458571 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #16320 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1015 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)(3,7)(4,9)(5,10,20,28)(6,14)(8,11)(12,22)(13,21,27,19)(15,23,30,32)(16,24)(17,25)(18,29,31,26);(1,3,7,2,8,14,6,11)(4,9,22,24,17,25,16,12)(5,13,19,10,20,27,21,28)(15,23,18,26,30,32,31,29);(1,4)(2,9)(3,12)(5,18,20,31)(6,16)(7,22)(8,17)(10,26,28,29)(11,25)(13,23,27,32)(14,24)(15,19,30,21);(1,5)(2,10)(3,13)(4,15)(6,21)(7,19)(8,20)(9,23)(11,28)(12,29)(14,27)(16,31)(17,30)(18,22)(24,26)(25,32)Polycyclic presentation:
2^9|a^2=f,b^2=efi,c^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,h^b=hi,d^c=de,g^c=gi,e^d=ei,f^d=fi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72237063, #72237070, #72237084, … |
| maximal normal subgroup of | #72237063, #72237070, #72237084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 365768);