Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #454041 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2668 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3019 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2223 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)(7,16)(8,21)(9,26)(10,14)(11,15)(12,18)(13,24)(17,27)(19,28)(20,32)(22,29)(23,31)(25,30)(33,34)(35,36);(5,7)(6,12)(8,22)(9,19)(10,18)(11,20)(13,29)(14,32)(15,28)(16,26)(17,23)(21,27)(24,35)(25,34)(30,36)(31,33);(2,4)(5,8,11,25)(6,13,15,31)(7,17,20,35)(9,23,10,24)(12,27,28,36)(14,30,26,21)(16,33,32,29)(18,22,19,34);(8,23)(9,10)(13,30)(14,26)(16,32)(17,22)(18,19)(21,31)(24,25)(27,33)(29,36)(34,35)Polycyclic presentation:
2^9|c^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi,g^d=gi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72256511, #72256515, #72256520 |
| maximal normal subgroup of | #72256511, #72256515, #72256520 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 361239);