Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996607 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#120 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2677 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)(7,9)(8,10)(11,12)(13,14,16,20,17,21,25,18)(15,22,23,26,24,19,27,28);(5,6)(7,11)(8,10)(9,12)(14,18)(15,23)(16,25)(20,21)(24,27)(26,28);(2,4)(5,7)(6,9)(8,12)(10,11);(7,12)(9,11)(13,15,16,23,17,24,25,27)(14,19,20,28,21,22,18,26);(6,10)(9,11)(15,24)(19,22)(23,27)(26,28)Polycyclic presentation:
2^9|a^2=f,d^2=f,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72282666, #72283423, #72283610, … |
| maximal normal subgroup of | #72282666, #72283423, #72283610, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6903931);