Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C23×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C22 | C22×C4×D4 |
| prime radical | #60955 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1550 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,8)(3,9,12,16)(4,10,5,14)(7,13,15,11)(17,18,21,26,22,25,32,38)(19,23,29,34,28,35,41,45)(20,24,31,37,30,36,42,46)(27,33,40,44,39,43,47,48);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,14)(10,16)(17,19)(18,23)(20,27)(21,29)(22,28)(24,33)(25,35)(26,34)(30,39)(31,40)(32,41)(36,43)(37,44)(38,45)(42,47)(46,48);(2,8)(7,15)(9,16)(10,14)(17,20,21,31,22,30,32,42)(18,24,26,37,25,36,38,46)(19,27,29,40,28,39,41,47)(23,33,34,44,35,43,45,48);(1,3)(2,9)(4,13)(5,11)(6,12)(7,14)(8,16)(10,15)(20,30)(24,36)(27,39)(31,42)(33,43)(37,46)(40,47)(44,48);(1,4,6,5)(2,10,8,14)(3,11,12,13)(7,16,15,9)(18,25)(19,28)(24,36)(26,38)(27,39)(29,41)(37,46)(40,47)Polycyclic presentation:
2^9|a^2=g,c^2=gh,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,e^a=ei,f^a=fh,e^b=ei,f^b=fh,d^c=di,e^d=eh92 conjugacy classes — view the full character table.
| maximal subgroup of | #72287585, #72287820, #72319568, … |
| maximal normal subgroup of | #72287585, #72287820, #72319568, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252880);