Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7500669 |
| 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 |
| prime radical | C22×#459 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2372 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)(7,9)(8,10)(11,12)(13,14,16,20,17,21,26,19)(15,22,24,27,23,28,25,18);(1,2)(3,4)(5,7,9,6)(8,11,12,10);(2,4)(5,8,9,12)(6,10,7,11)(13,15)(14,18)(16,25)(17,23)(19,22)(20,28)(21,27)(24,26);(6,7)(10,11)(14,19)(16,26)(18,22)(20,21)(24,25)(27,28);(15,23)(18,27)(22,28)(24,25)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df,e^c=ei,g^c=gi,g^d=gi74 conjugacy classes — view the full character table.
| maximal subgroup of | #72280025, #72280303, #72310835, … |
| maximal normal subgroup of | #72280025, #72280303, #72310835, … |
| maximal quotient of | #72313049, #72313224, #72313672 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410901);