Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2735949 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3480 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)(5,6,10,9)(7,12,8,11)(13,14,17,20)(15,19,16,18)(22,26)(27,28);(5,7)(6,11)(8,10)(9,12)(15,16)(18,19)(21,22,25,26)(23,27,24,28);(7,8)(11,12)(13,15,17,16)(14,18,20,19)(21,23,25,24)(22,27,26,28);(1,2)(3,4)(5,8,10,7)(6,11,9,12)(13,16,17,15)(14,19,20,18)(21,22)(23,28)(24,27)(25,26);(5,9,10,6)(7,12,8,11)(13,14)(15,18)(16,19)(17,20)(21,24,25,23)(22,28,26,27)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,c^2=fh,d^2=h,e^2=f,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.
| maximal subgroup of | #72297822, #72297828, #72297851 |
| maximal normal subgroup of | #72297822, #72297828, #72297851 |
| maximal quotient of | #72298297 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2644820);