Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493657 |
| 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 | #63024 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2242 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,13,3,11)(4,16,14,27,18,9,21,26)(5,8,15,24,19,10,20,23)(17,25,30,32,29,22,28,31);(2,8)(3,6)(5,19)(9,22)(10,13)(11,23)(12,24)(14,21)(16,25)(17,29)(26,31)(27,32);(1,3)(4,14)(5,15)(6,7)(8,10)(11,12)(17,28)(18,21)(19,20)(22,25)(26,27)(29,30);(1,4)(2,9)(3,14)(5,17)(6,21)(7,18)(8,22)(10,25)(11,26)(12,27)(13,16)(15,28)(19,29)(20,30)(23,31)(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,c^b=ci,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=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72261065, #72261107, #72261130 |
| maximal normal subgroup of | #72261065, #72261107, #72261130 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400859);