Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392237 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14681 |
| 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 | |
3491 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)(3,11,14,7)(4,15,17,30)(5,10)(6,9)(8,16,21,31)(12,25)(13,24,28,20)(18,22)(19,26)(23,32)(27,29);(1,3)(2,7)(4,12)(5,13)(6,14)(8,19)(9,11)(10,20)(15,29)(16,26)(17,27)(18,28)(21,23)(22,24)(25,30)(31,32);(1,4)(2,8,9,21)(3,12)(5,16)(6,17)(7,19,11,23)(10,15,22,30)(13,26)(14,27)(18,31)(20,29,24,25)(28,32);(5,18)(10,22)(13,28)(15,30)(16,31)(20,24)(25,29)(26,32)Polycyclic presentation:
2^9|a^2=h,c^2=e,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,h^b=hi,g^c=gi,h^c=hi,f^d=fi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72251414, #72251416, #72251731, … |
| maximal normal subgroup of | #72251414, #72251416, #72251731, … |
| maximal quotient of | #72255852 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299433);