Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| automorphism | Aut(G) order 1048576 |
| 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 | C2×#2410 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3750 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14)(15,18)(16,20)(17,19)(21,22,25,26)(23,27,28,24);(5,6)(7,9)(8,10)(11,12)(13,15,17,20)(14,18,19,16)(21,23)(22,24)(25,28)(26,27);(7,12)(9,11)(13,15)(14,16)(17,20)(18,19)(21,24,25,27)(22,23,26,28);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,28);(5,7)(6,9)(8,12)(10,11)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72293912, C2×#4599121, #72336514, … |
| maximal normal subgroup of | #72293912, C2×#4599121, #72336514, … |
| maximal quotient of | #72293981, C2×#4599121 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4506317);