Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #395154 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
962 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2,3,4)(5,6)(7,12)(8,11)(9,10)(21,22)(23,34)(24,28)(25,32)(26,31)(27,33)(29,35)(30,36);(2,4)(6,10)(11,12)(13,14,16,18)(15,19,20,17)(21,23,26,35,27,36,25,24)(22,28,32,30,33,29,31,34);(5,7,9,8)(6,11,10,12)(13,15,16,20)(14,17,18,19)(21,23,25,24,27,36,26,35)(22,29,31,30,33,28,32,34);(5,8,9,7)(6,11,10,12)(21,24,25,36,27,35,26,23)(22,30,31,28,33,34,32,29)Polycyclic presentation:
2^9|a^2=ef,b^2=gh,c^2=fg,d^2=fi,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^c=di80 conjugacy classes — view the full character table.
| maximal subgroup of | #72308282, #72308305, #72308378, … |
| maximal normal subgroup of | #72308282, #72308305, #72308378, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 318344);