Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6342728 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×C22⋊C4 |
| commutator | C23 | C24×C4 |
| Frattini, cosocle | C24 | C25 |
| prime radical | C2×#3500 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×#124 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
4790 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(5,6)(7,11)(8,18,20,12)(9,13)(10,14)(15,29,31,23)(16,24)(17,25)(19,32,34,26)(21,27)(22,28)(30,36,33,35);(1,2)(3,4)(5,7,10,17)(6,11,14,25)(8,15,20,31)(9,21,22,16)(12,23,18,29)(13,27,28,24)(19,33,34,30)(26,36,32,35);(5,8)(6,12)(7,15)(9,19)(10,20)(11,23)(13,26)(14,18)(16,30)(17,31)(21,33)(22,34)(24,35)(25,29)(27,36)(28,32);(2,4)(9,22)(13,28)(16,21)(19,34)(24,27)(26,32)(30,33)(35,36);(5,9,10,22)(6,13,14,28)(7,16,17,21)(8,19,20,34)(11,24,25,27)(12,26,18,32)(15,30,31,33)(23,35,29,36)Polycyclic presentation:
2^9|a^2=h,b^2=i,e^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=df,e^b=ei,h^c=hi,e^d=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72306245, #72306265, #72306269, … |
| maximal normal subgroup of | #72306245, #72306265, #72306269, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6286212);