Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #465214 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2535 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#2008 | C2 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1133 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)(6,11)(8,19)(9,22)(12,27)(13,29)(14,26)(15,24)(16,20)(17,23)(21,33)(28,31)(32,35);(5,6,9,11,10,14,22,26)(7,15,23,27,18,12,17,24)(8,13,19,25,21,29,33,36)(16,31,35,30,32,28,20,34);(5,7)(6,12,14,15)(8,20)(9,23)(10,18)(11,24,26,27)(13,30,29,34)(16,19)(17,22)(21,35)(25,28,36,31)(32,33);(2,4)(5,8,9,19,10,21,22,33)(6,13,11,25,14,29,26,36)(7,16,17,20,18,32,23,35)(12,28,27,30,15,31,24,34)Polycyclic presentation:
2^9|b^2=eg,c^2=f,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,f^b=fi,d^c=de,e^c=ei,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 372410);