Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24408 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2410 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C2 | #13995 |
| prime radical | #60398 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2238 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,12,7,10,6,11)(3,13,17,14,18,8,16,24)(4,19,20,25,21,9,22,26)(15,23,30,28,31,27,29,32);(1,3,5,16,7,18,6,17)(2,8,12,14,10,13,11,24)(4,15,22,30,21,31,20,29)(9,23,25,28,19,27,26,32);(1,2)(3,14)(4,9)(5,12)(6,11)(7,10)(8,17)(13,16)(15,28)(18,24)(19,21)(20,26)(22,25)(23,29)(27,30)(31,32);(1,4)(2,9)(3,15)(5,22)(6,20)(7,21)(8,23)(10,19)(11,26)(12,25)(13,27)(14,28)(16,30)(17,29)(18,31)(24,32)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,f^2=i,g^2=i,h^2=i,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,f^c=fi,g^c=gi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72319399, #72319406, #72319444, … |
| maximal normal subgroup of | #72319399, #72319406, #72319444, … |
| maximal quotient of | #72343051 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273841);