Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| 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 | |
3799 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)(13,14,16,18)(15,19,20,17)(21,22,26,28)(23,24,25,27);(5,7)(6,9)(8,12)(10,11)(13,14)(15,19)(16,18)(17,20)(21,23,26,25)(22,27,28,24);(6,10)(9,11)(15,20)(17,19)(21,24,26,27)(22,23,28,25);(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,15)(14,17)(16,20)(18,19)(21,24)(22,23)(25,28)(26,27);(5,7)(6,11)(8,12)(9,10)(13,14)(15,19)(16,18)(17,20)(21,25,26,23)(22,27,28,24)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72235982, #72235995, #72235996, … |
| maximal normal subgroup of | #72235982, #72235995, #72235996, … |
| maximal quotient of | #72240456, #72332991 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2672218);