Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #484139 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33719 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1770 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
674 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 36 points:
(2,4)(5,6,10,15,11,16,24,26,9,13,8,21,23,29,22,12)(7,17,18,27,20,28,35,31,19,33,25,34,36,14,30,32);(6,12,13,26)(7,18,19,25)(8,22)(10,24)(11,23)(15,16,21,29)(17,33)(20,30,36,35)(27,32)(31,34);(1,2,3,4)(6,13)(12,26)(14,28)(15,21)(16,29)(17,33)(27,34)(31,32);(5,7,8,18,11,20,22,35,9,19,10,25,23,36,24,30)(6,14,15,31,16,17,26,34,13,28,21,32,29,33,12,27)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=fi,d^2=e,e^2=h,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,d^b=de,e^b=eh,g^b=gh,h^b=hi,g^d=gi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72250153, #72250863, #72250867, … |
| maximal normal subgroup of | #72250153, #72250863, #72250867, … |
| maximal quotient of | #72287611, #72287705, #72287762, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 405776);