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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2403 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4658 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)(15,20)(16,17)(18,19)(22,24)(23,28);(7,12)(9,11)(14,19)(15,16)(21,22,25,24)(23,27,28,26);(5,7)(6,9)(8,12)(10,11)(13,15,18,16)(14,20,19,17)(21,23,25,28)(22,26,24,27);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,16,18,15)(14,17,19,20)(21,24,25,22)(23,27,28,26);(5,6)(7,9)(8,10)(11,12)(13,17)(14,16)(15,19)(18,20)(21,23)(22,27)(24,26)(25,28)Polycyclic presentation:
2^9|a^2=fh,b^2=fgh,c^2=f,d^2=f,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 | C2×#3335868, #72293937, #72300712, … |
| maximal normal subgroup of | C2×#3335868, #72293937, #72300712, … |
| maximal quotient of | #72293743, C2×#3335868, #72329890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3243064);