Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23998 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2404 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13967 |
| prime radical | C2×#3018 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2341 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,8,3,6,11,7,12)(4,13,18,9,19,15,17,25)(5,20,22,26,21,10,23,14)(16,27,29,24,30,28,31,32);(1,3,8,11,6,12,7,2)(4,15,18,25,19,13,17,9)(5,14,22,20,21,26,23,10)(16,28,29,32,30,27,31,24);(1,4)(2,9)(3,13)(5,16)(6,19)(7,18)(8,17)(10,24)(11,25)(12,15)(14,27)(20,32)(21,30)(22,31)(23,29)(26,28);(1,5)(2,10)(3,14)(4,16)(6,21)(7,23)(8,22)(9,24)(11,20)(12,26)(13,27)(15,28)(17,31)(18,29)(19,30)(25,32)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,g^b=gi,e^c=ei,f^c=fi,h^c=hi,f^d=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72215906, #72219687, #72219688, … |
| maximal normal subgroup of | #72215906, #72219687, #72219688, … |
| maximal quotient of | C2×#364512 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 271708);