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 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3739 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)(5,6,9,10)(7,11,12,8)(13,14)(15,18)(16,17)(19,20)(22,24)(23,27);(5,7)(6,8)(9,12)(10,11)(14,17)(18,19)(23,27)(26,28);(6,10)(8,11)(13,14)(15,19)(16,17)(18,20)(21,22,25,24)(23,26,27,28);(1,2)(3,4)(5,7)(6,11)(8,10)(9,12)(13,14)(15,19)(16,17)(18,20)(21,23,25,27)(22,26,24,28);(5,8,9,11)(6,12,10,7)(13,15,16,20)(14,18,17,19)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^9|a^2=fh,c^2=fgh,d^2=fgh,e^2=gh,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 | #72329305, #72329353, #72329369, … |
| maximal normal subgroup of | #72329305, #72329353, #72329369, … |
| maximal quotient of | #72330316, #72332818 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3270518);