Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #359903 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13869 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2095 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,8,5,11)(3,13,16,22,17,23,14,9)(4,18,19,26,20,10,21,25)(15,29,28,32,31,27,30,24);(2,8)(3,14)(5,6)(9,23)(10,18)(13,22)(15,28)(16,17)(19,21)(24,29)(27,32)(30,31);(1,3)(2,9,8,22)(4,15)(5,16)(6,14)(7,17)(10,24,18,32)(11,13,12,23)(19,30)(20,31)(21,28)(25,27,26,29);(1,4)(2,10)(3,15)(5,19)(6,21)(7,20)(8,18)(9,24)(11,25)(12,26)(13,27)(14,28)(16,30)(17,31)(22,32)(23,29)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi,g^b=gi,h^b=hi,g^c=gi,h^c=hi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72319119, #72319375, #72319460, … |
| maximal normal subgroup of | #72319119, #72319375, #72319460, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 267101);