Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| 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 | |
2172 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,10,8)(7,11,9,12)(13,14,18,16)(15,19,17,20)(21,22,26,24)(23,25,27,28);(6,8)(11,12)(13,15,18,17)(14,19,16,20)(22,24)(23,27);(5,7,10,9)(6,11,8,12)(13,15,18,17)(14,20,16,19)(21,23,26,27)(22,25,24,28);(1,2)(3,4)(5,8,10,6)(7,11,9,12)(13,16,18,14)(15,19,17,20)(21,24,26,22)(23,25,27,28);(5,9,10,7)(6,12,8,11)(13,17,18,15)(14,19,16,20)(21,25)(22,27)(23,24)(26,28)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fg,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,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, 2645558);