Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 | |
4427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6,8,10)(7,9,12,11)(14,17)(18,19)(21,22,25,27)(23,28,26,24);(5,6)(7,9)(8,10)(11,12)(13,14)(15,19)(16,17)(18,20)(21,23,25,26)(22,24,27,28);(13,15)(14,18)(16,20)(17,19)(21,22)(23,28)(24,26)(25,27);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,20)(17,18)(21,22)(23,24)(25,27)(26,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=i,b^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72297872, #72297885, #72297890, … |
| maximal normal subgroup of | #72297872, #72297885, #72297890, … |
| maximal quotient of | #72298116, #72329959, #72330622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4327800);