Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1393169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2075 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,8)(7,11)(21,22,26,29)(23,27,31,25)(24,32,35,28)(30,36,33,34);(14,17)(18,20)(21,23,26,31)(22,25,29,27)(24,30,35,33)(28,36,32,34);(5,6,9,8)(7,10,11,12)(13,14,16,17)(15,18,19,20)(21,23)(22,27)(24,33)(25,29)(26,31)(28,36)(30,35)(32,34);(1,2,3,4)(5,7)(6,10)(8,12)(9,11)(21,24,26,35)(22,28,29,32)(23,30,31,33)(25,36,27,34);(5,8,9,6)(7,12,11,10)(13,15,16,19)(14,18,17,20)(21,25,26,27)(22,23,29,31)(24,34,35,36)(28,33,32,30)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=fgh,d^2=gi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^c=df,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1317796);