Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #200412 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #2617 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10)(11,19,22,33)(12,23,25,16)(13,18)(14,17)(15,21,30,26)(20,35)(24,34,27,31)(28,32)(29,36)(37,40)(38,39);(2,6,8,7)(9,11,13,26,14,22,28,21)(10,15,18,33,17,30,32,19)(12,20,27,36,25,37,24,38)(16,29,34,40,23,39,31,35);(9,12)(10,16)(11,20)(13,27)(14,25)(15,29)(17,23)(18,34)(19,35)(21,38)(22,37)(24,28)(26,36)(30,39)(31,32)(33,40);(10,17)(15,30)(16,23)(18,32)(19,33)(29,39)(31,34)(35,40)Polycyclic presentation:
2^9|a^2=g,b^2=eg,e^2=hi,g^2=h,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,e^b=ei,f^b=fi,g^b=gi,e^c=ei,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72340417, #72340419, #72340425, … |
| maximal normal subgroup of | #72340417, #72340419, #72340425, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 107656);