Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395173 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2408 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1846 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)(3,4)(5,6)(7,17)(8,22)(9,16)(10,14)(11,15)(12,19)(13,24)(18,27)(20,28)(21,26)(23,30)(25,29)(31,34)(32,35)(33,36);(5,7,9,19,11,20,10,21)(6,12,14,17,15,26,16,28)(8,18,23,32,25,33,24,34)(13,27,29,31,30,36,22,35);(12,26)(17,28)(27,36)(31,35);(2,4)(5,8,9,23,11,25,10,24)(6,13,14,29,15,30,16,22)(7,18,19,32,20,33,21,34)(12,27,17,31,26,36,28,35)Polycyclic presentation:
2^9|b^2=e,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,f^b=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72262367, #72262403, #72262411, … |
| maximal normal subgroup of | #72262367, #72262403, #72262411, … |
| maximal quotient of | #72314528 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 302369);