Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| 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 | |
4246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6)(7,11)(8,9)(10,12)(13,14,18,19)(15,20,16,17)(21,22,25,24)(23,27,28,26);(2,4)(5,7)(6,10)(8,11)(9,12)(14,19)(15,16)(21,23)(22,26)(24,27)(25,28);(1,2)(3,4)(13,15,18,16)(14,17,19,20)(21,22,25,24)(23,27,28,26);(5,8,9,6)(7,11,12,10)(13,16,18,15)(14,17,19,20)(21,24,25,22)(23,27,28,26);(5,7)(6,11)(8,10)(9,12)(13,17)(14,16)(15,19)(18,20)(21,23)(22,27)(24,26)(25,28)Polycyclic presentation:
2^9|a^2=fi,c^2=fi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72262655, #72262665, #72262675, … |
| maximal normal subgroup of | #72262655, #72262665, #72262675, … |
| maximal quotient of | #72329852, #72330166, #72330561, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2540438);