Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6431707 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3318 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14,17,21,18,16,26,20)(15,22,25,27,23,28,24,19);(1,2)(3,4)(5,7,8,12)(6,9,10,11)(13,15,18,23)(14,19,16,27)(17,24,26,25)(20,22,21,28);(2,4)(13,16)(14,18)(15,19)(17,21)(20,26)(22,24)(23,27)(25,28);(15,23)(19,27)(22,28)(24,25);(13,17,18,26)(14,20,16,21)(15,24,23,25)(19,22,27,28)Polycyclic presentation:
2^9|a^2=f,b^2=fg,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,f^b=fi,g^b=gi,f^c=fi,g^c=gi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72307408, #72307422, #72307532, … |
| maximal normal subgroup of | #72307408, #72307422, #72307532, … |
| maximal quotient of | #72316033 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339616);