Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61058 |
| 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 | #3061 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #2965 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
1531 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,6,9,5,8,14,16)(3,10,12,13,11,7,15,4)(17,18)(19,28)(20,26)(21,27)(22,35)(23,25)(24,30)(29,43)(31,39)(32,41)(33,40)(34,42)(36,37)(38,45)(44,47)(46,48);(1,3)(2,7)(4,9)(5,11)(6,12)(8,10)(13,16)(14,15)(17,19,23,31)(18,24,25,37)(20,29,33,46)(21,30,22,36)(26,38,40,47)(27,39,35,28)(32,44,34,45)(41,43,42,48);(1,4,6,10,5,13,14,7)(2,3,9,12,8,11,16,15)(18,25)(24,37)(26,40)(27,35)(28,39)(38,47)(41,42)(43,48);(3,11)(4,13)(7,10)(12,15)(17,20)(18,26)(19,29)(21,34)(22,32)(23,33)(24,38)(25,40)(27,42)(28,43)(30,45)(31,46)(35,41)(36,44)(37,47)(39,48);(17,21,23,22)(18,27,25,35)(19,30,31,36)(20,32,33,34)(24,39,37,28)(26,41,40,42)(29,44,46,45)(38,43,47,48)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=g,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=ch,f^a=fh,d^b=di,f^b=fh,d^c=di,e^d=eh104 conjugacy classes — view the full character table.
| maximal subgroup of | #72322152, #72322173, #72322235, … |
| maximal normal subgroup of | #72322152, #72322173, #72322235, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6252371);