Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #495899 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #21526 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33618 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2875 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
625 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,8,11,23)(3,13)(4,16)(5,6)(9,24,27,32)(10,21,12,22)(14,29)(15,19)(17,18)(20,25,31,26);(1,2,4,8,6,12,16,21,7,11,17,23,5,10,18,22)(3,9,14,20,13,26,28,24,15,27,29,31,19,25,30,32);(1,3)(2,9)(4,14)(5,19)(6,13)(7,15)(8,20)(10,25)(11,27)(12,26)(16,28)(17,29)(18,30)(21,24)(22,32)(23,31);(2,10,11,12)(4,17)(8,21,23,22)(9,25,27,26)(14,29)(16,18)(20,24,31,32)(28,30)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=g,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,d^b=dh,g^b=gi,f^c=fi,e^d=ei65 conjugacy classes — view the full character table.
| maximal subgroup of | #72214039, #72237280, C2×#495899, … |
| maximal normal subgroup of | #72214039, #72237280, C2×#495899, … |
| maximal quotient of | #72216451, #72227176, #72227177, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 403095);