Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #148578 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #11196 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C22×H(5,𝔽2) | C22 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ C22×H(5,𝔽2) ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1367 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,15,10)(4,16)(5,9)(6,11)(7,14,23,20)(8,13,26,17)(18,22)(19,28)(21,27)(24,32)(25,29,30,31);(1,3)(2,7)(4,17)(5,20)(6,15)(8,24)(9,10)(11,23)(12,27)(13,19)(14,21)(16,30)(18,29)(22,26)(25,28)(31,32);(1,4,5,18,6,19,21,32)(2,8,9,25,11,26,27,30)(3,13,14,29,15,17,20,31)(7,22,12,28,23,24,10,16)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=i,b^a=bd,c^a=ce,e^a=eh,h^a=hi,c^b=cf,d^b=di,g^b=gi,f^c=fi,g^c=gh,g^e=gi53 conjugacy classes — view the full character table.
| maximal subgroup of | #72267966 |
| maximal normal subgroup of | #72267966 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55774);