Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #392712 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #14694 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
3844 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)(3,8,14,24)(4,17,18,26)(5,11)(6,12)(7,10)(9,19,25,20)(13,27)(15,22,16,23)(21,30)(28,32)(29,31);(1,3)(2,8,10,24)(4,13)(5,16)(6,15)(7,14)(9,21,25,32)(11,23,12,22)(17,31,26,27)(18,29)(19,30)(20,28);(1,4)(2,9)(3,13)(5,20)(6,19)(7,18)(8,21)(10,25)(11,26)(12,17)(14,29)(15,30)(16,28)(22,31)(23,27)(24,32);(1,4)(2,9)(3,13)(5,19)(6,20)(7,18)(8,21)(10,25)(11,17)(12,26)(14,29)(15,28)(16,30)(22,27)(23,31)(24,32)Polycyclic presentation:
2^9|a^2=h,b^2=eh,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,f^c=fi,h^c=hi,e^d=ei,g^d=gi,h^d=hi89 conjugacy classes — view the full character table.
| maximal subgroup of | #72235139, #72238712, #72238722, … |
| maximal normal subgroup of | #72235139, #72238712, #72238722, … |
| maximal quotient of | #72235178, #72286507, #72286562 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299908);