Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #493483 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #20585 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #62876 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23 ◁ #3487 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
2268 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,12,7,13,3,9)(4,10,18,25,17,27,14,24)(5,8,15,21,19,11,20,23)(16,22,28,31,29,26,30,32);(2,8)(3,6)(5,19)(9,23)(10,22)(11,13)(12,21)(14,18)(16,29)(24,32)(25,31)(26,27);(1,3,7,6)(2,9,13,12)(4,14,17,18)(5,15,19,20)(8,21,11,23)(10,25,27,24)(16,28,29,30)(22,32,26,31);(1,4)(2,10)(3,14)(5,16)(6,18)(7,17)(8,22)(9,24)(11,26)(12,25)(13,27)(15,28)(19,29)(20,30)(21,31)(23,32)Polycyclic presentation:
2^9|a^2=g,c^2=i,e^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,c^b=ci,e^b=ei,g^b=ghi,d^c=dh,f^d=fi,g^e=gi59 conjugacy classes — view the full character table.
| maximal subgroup of | #72261130, #72261176 |
| maximal normal subgroup of | #72261130, #72261176 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400686);