Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #481393 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#102 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C4×C8 | C24 |
| prime radical | #60695 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ C4×#131 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
733 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2,4,8,5,7,11,14)(3,6,10,13,9,12,15,16)(17,18,21,27,22,25,36,28)(19,24,31,41,29,43,45,42)(20,32,34,26,35,23,33,38)(30,39,46,40,44,37,47,48);(1,3)(2,6)(4,10)(5,9)(7,12)(8,13)(11,15)(14,16)(18,23)(19,29)(20,33)(21,36)(24,39)(25,32)(26,27)(28,38)(30,46)(34,35)(37,43)(40,42)(41,48)(44,47);(3,9)(6,12)(10,15)(13,16)(17,19)(18,24)(20,30)(21,31)(22,29)(23,37)(25,43)(26,40)(27,41)(28,42)(32,39)(33,47)(34,46)(35,44)(36,45)(38,48);(2,7)(6,12)(8,14)(13,16)(18,25)(19,29)(23,32)(26,38)(27,28)(30,44)(31,45)(46,47)Polycyclic presentation:
2^9|a^2=f,e^2=g,f^2=h,g^2=i,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=ch,e^b=egi,f^b=fi,g^b=gi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286070, #72286279, #72286755, … |
| maximal normal subgroup of | #72286070, #72286279, #72286755, … |
| maximal quotient of | #72343327, #72343371, #72343396, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388589);