Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #417589 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C4×#95 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1154 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,12)(9,11)(13,14,16,18)(15,17,19,20)(21,22,26,28)(23,31,32,29)(24,27,34,36)(25,35,33,30);(5,7,9,8)(6,10,11,12)(13,15,16,19)(14,17,18,20)(21,23)(22,27)(24,33)(25,34)(26,32)(28,36)(29,30)(31,35);(5,8,9,7)(6,10,11,12)(15,19)(17,20)(22,28)(23,32)(24,34)(30,35);(2,4)(21,24,33,23,26,34,25,32)(22,29,35,36,28,31,30,27)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=fi,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 348871);