Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×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 | |
5239 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,12)(13,15)(14,16)(17,18,21,24)(19,20,23,22);(5,6,7,8)(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,24)(21,23);(5,6)(7,8)(10,15)(12,14)(19,23)(20,22);(1,2)(3,4)(9,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,24)(22,23);(9,12)(10,16)(11,15)(13,14)(17,20)(18,23)(19,24)(21,22)Polycyclic presentation:
2^9|a^2=fg,b^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72293937, C2×#3334864, #72329800, … |
| maximal normal subgroup of | #72293937, C2×#3334864, #72329800, … |
| maximal quotient of | #72293970, C2×#3334864, #72330381, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3242060);