Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7514196 |
| 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 | C2×#3318 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#123 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2245 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(5,6)(7,12)(8,10)(9,11)(13,14)(15,20)(16,19)(17,18);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(13,14,16,19)(15,20,17,18);(14,18)(15,17)(19,20);(2,4)(5,8,9,7)(6,10,11,12)(13,15)(16,17)(18,20);(6,11)(10,12)(13,15,16,17)(14,18,19,20)Polycyclic presentation:
2^9|b^2=f,d^2=fi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,h^b=hi,d^c=di,e^c=ei,g^c=gi,h^c=hi,e^d=ei,g^d=gi,h^d=hi77 conjugacy classes — view the full character table.
| maximal subgroup of | #72249685, #72282890, #72283660, … |
| maximal normal subgroup of | #72249685, #72282890, #72283660, … |
| maximal quotient of | #72249733, #72305538, #72305866, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7425331);