Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C2×#2613 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3885 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(5,6,8,10)(7,9,12,11)(13,14,16,19)(15,17,20,18)(21,22,25,27)(23,24,28,26);(5,6)(7,9)(8,10)(11,12)(13,14)(15,18)(16,19)(17,20)(21,23)(22,26)(24,27)(25,28);(1,2,4,3)(13,15)(14,17)(16,20)(18,19)(21,23)(22,26)(24,27)(25,28);(1,3,4,2)(5,6)(7,11)(8,10)(9,12)(13,15)(14,18)(16,20)(17,19)(21,24,25,26)(22,23,27,28);(5,7)(6,9)(8,12)(10,11)(13,14)(15,17)(16,19)(18,20)(21,23)(22,24)(25,28)(26,27)Polycyclic presentation:
2^9|a^2=f,c^2=fi,d^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 | #72326416, #72326421, #72326438, … |
| maximal normal subgroup of | #72326416, #72326421, #72326438, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4337605);