Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C24 | C25 |
| commutator | C23 | C24×C4 |
| prime radical | C2×#2403 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,4,3)(5,6,9,7)(8,11,12,10)(14,15)(18,19)(21,22,26,28)(23,25,24,27);(1,3,4,2)(6,7)(8,12)(13,14,17,15)(16,19,20,18)(21,23)(22,27)(24,26)(25,28);(6,7)(10,11)(14,15)(18,19)(22,28)(23,24);(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18)(21,24,26,23)(22,25,28,27);(5,8)(6,10)(7,11)(9,12)(13,16)(14,18)(15,19)(17,20)(21,25)(22,23)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=i,b^2=fi,d^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=eg,c^b=ch,d^b=dg,e^b=ef,d^c=df92 conjugacy classes — view the full character table.
| maximal subgroup of | #72263270, #72263290, #72263303, … |
| maximal normal subgroup of | #72263270, #72263290, #72263303, … |
| maximal quotient of | #72328515, #72330306, #72332857, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 494536);