Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #366633 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #13995 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C25 | C24 |
| prime radical | C2×#3026 | C2 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C2 ▷ C1 | |
2440 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)(3,11,14,7)(4,15,18,8)(5,19,20,9)(6,10)(12,21)(13,22)(16,24)(17,28)(23,30)(25,27)(26,31,29,32);(1,3)(2,7)(4,16,18,12)(5,13)(6,14)(8,21,15,24)(9,22)(10,11)(17,29,30,26)(19,25)(20,27)(23,31,28,32);(1,4)(2,8)(3,12)(5,17)(6,18)(7,21)(9,23)(10,15)(11,24)(13,26)(14,16)(19,28)(20,30)(22,31)(25,32)(27,29);(1,5)(2,9)(3,13)(4,17)(6,20)(7,22)(8,23)(10,19)(11,25)(12,26)(14,27)(15,28)(16,29)(18,30)(21,31)(24,32)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,g^b=gi,h^b=hi,e^c=ei,h^c=hi,f^d=fi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72210507, #72237762, #72237764, … |
| maximal normal subgroup of | #72210507, #72237762, #72237764, … |
| maximal quotient of | #72240598, #72258150, C2×#366633 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273829);