Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| 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 | |
4269 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)(5,6,8,9)(7,11,12,10)(14,17)(18,20)(22,26)(23,28);(6,9)(7,12)(13,14,16,17)(15,18,19,20)(21,22,25,26)(23,24,28,27);(6,9)(10,11)(15,19)(18,20)(22,26)(24,27);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,14)(15,20)(16,17)(18,19)(21,23)(22,27)(24,26)(25,28);(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,19)(17,20)(21,24,25,27)(22,28,26,23)Polycyclic presentation:
2^9|a^2=g,b^2=fg,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232298, #72232318, #72232334, … |
| maximal normal subgroup of | #72232298, #72232318, #72232334, … |
| maximal quotient of | #72329026, #72331338, C2×#4045313 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3952509);