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 | C2×#41 | C24 |
| prime radical | #63024 | C2 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C2×2−1+4 ◁ G | |
| LECS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
2245 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)(3,8,5,11)(4,10,18,27)(6,12)(7,13)(9,15,23,19)(14,22)(16,25)(17,26,30,31)(20,21)(24,29)(28,32);(2,8)(3,5)(9,12)(10,22)(11,13)(14,16)(15,19)(21,23)(24,26)(25,27)(28,29)(31,32);(1,3)(2,9)(4,14)(5,7)(6,19)(8,21)(10,24)(11,12)(13,23)(15,20)(16,18)(17,29)(22,31)(25,26)(27,32)(28,30);(1,4)(2,10)(3,14)(5,16)(6,17)(7,18)(8,22)(9,24)(11,25)(12,26)(13,27)(15,28)(19,29)(20,30)(21,31)(23,32)Polycyclic presentation:
2^9|a^2=g,e^2=i,f^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 | #72261708 |
| maximal normal subgroup of | #72261708 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400853);