Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493653 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20609 |
| commutator | C2×Q8 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #63045 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2297 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,9,7,13,5,12)(3,14,16,25,17,10,21,27)(4,8,20,22,18,11,19,24)(15,26,29,31,28,23,30,32);(2,8)(4,18)(5,6)(9,22)(10,23)(11,13)(12,24)(14,26)(15,28)(16,21)(25,31)(27,32);(2,9)(5,6)(8,22)(10,25)(11,24)(12,13)(14,27)(16,21)(19,20)(23,31)(26,32)(29,30);(1,3)(2,10)(4,15)(5,16)(6,21)(7,17)(8,23)(9,25)(11,26)(12,27)(13,14)(18,28)(19,29)(20,30)(22,31)(24,32)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=eh,h^a=hi,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,g^d=gi,f^e=fi,g^e=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72260900, #72261065, #72261130 |
| maximal normal subgroup of | #72260900, #72261065, #72261130 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400857);