Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
2152 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)(5,6,9,12)(7,11,8,10)(15,19)(16,18)(21,22,26,27)(23,24,30,28)(25,29,34,35)(31,32,36,33);(5,6,9,12)(7,10,8,11)(13,14,17,20)(15,16,19,18)(22,27)(23,30)(29,35)(31,36);(5,7,9,8)(6,10,12,11)(13,15,17,19)(14,18,20,16)(21,23,26,30)(22,28,27,24)(25,31,34,36)(29,33,35,32);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,14)(15,18)(16,19)(17,20)(21,24,26,28)(22,23,27,30)(25,33,34,32)(29,36,35,31);(5,8,9,7)(6,11,12,10)(13,16,17,18)(14,19,20,15)(21,25)(22,29)(23,31)(24,32)(26,34)(27,35)(28,33)(30,36)Polycyclic presentation:
2^9|a^2=fgh,b^2=fg,c^2=g,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2668324);