Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483993 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #2218 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1450 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
405 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,5,9,6,3,10,8)(4,7,12,15,13,11,16,14)(17,18,21,28,23,26,22,40)(19,29,32,25,30,45,33,46)(20,34,37,42,39,48,38,27)(24,41,43,31,44,35,47,36);(1,3)(2,6)(4,11)(5,8)(7,13)(9,10)(12,14)(15,16)(17,19)(18,24)(20,35)(21,33)(22,32)(23,30)(25,42)(26,44)(27,46)(28,47)(29,48)(31,37)(34,45)(36,38)(39,41)(40,43);(1,4,10,16,6,13,5,12)(2,7,8,14,3,11,9,15)(17,19)(18,25,26,46)(20,36)(21,32)(22,33)(23,30)(24,42,44,27)(28,45,40,29)(31,39)(34,43,48,47)(35,38)(37,41);(2,3)(7,11)(8,9)(14,15)(18,26)(19,30)(27,42)(28,40)(31,36)(32,33)(34,48)(35,41)Polycyclic presentation:
2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=dh,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fi,g^b=gi,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286070, #72286149, #72286247, … |
| maximal normal subgroup of | #72286070, #72286149, #72286247, … |
| maximal quotient of | #72343441, #72343467, #72343496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 391189);