Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493464 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | C2×#1782 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #62855 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #3428 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2249 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,13,4,10)(3,14,18,24,17,9,15,26)(5,8,19,23,20,11,21,27)(16,25,28,32,29,22,30,31);(2,8)(4,6)(5,20)(9,22)(10,27)(11,13)(12,23)(14,25)(15,18)(16,29)(24,32)(26,31);(1,3)(2,9,13,14)(4,15)(5,16)(6,18)(7,17)(8,22,11,25)(10,24,12,26)(19,28)(20,29)(21,30)(23,31,27,32);(1,4,7,6)(2,10,13,12)(3,15,17,18)(5,19,20,21)(8,23,11,27)(9,24,14,26)(16,28,29,30)(22,31,25,32)Polycyclic presentation:
2^9|a^2=g,c^2=h,d^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,g^b=ghi,f^c=fi,g^e=gi65 conjugacy classes — view the full character table.
| maximal subgroup of | #72261089, #72261176 |
| maximal normal subgroup of | #72261089, #72261176 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400667);