Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23994 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2535 |
| prime radical | C25 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1012 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,19)(9,22)(10,14)(11,17)(12,20)(13,23)(16,29)(18,27)(21,33)(24,32)(25,31)(26,34)(28,30)(35,36);(1,3,4,2)(5,7)(6,11,14,24)(8,16)(9,17)(10,18)(12,25,26,29)(13,15,22,27)(19,28,33,35)(20,30)(21,31)(23,32)(34,36);(11,24)(15,27)(25,29)(28,35);(5,8,9,20,10,21,23,34)(6,12,13,19,14,26,22,33)(7,16,17,30,18,31,32,36)(11,25,15,28,24,29,27,35)Polycyclic presentation:
2^9|a^2=h,b^2=gh,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=gi,c^b=cg,f^b=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72250376, #72250386, #72250390, … |
| maximal normal subgroup of | #72250376, #72250386, #72250390, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 273639);