Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437510 |
| 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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 20 points:
(2,4)(5,6,8,10)(7,11,12,9)(13,14,19,20)(15,18,16,17);(1,2,3,4)(13,15,18,20,19,16,17,14);(5,7,8,12)(6,9,10,11)(13,15)(14,18)(16,19)(17,20);(13,16)(14,17)(15,19)(18,20);(13,17,19,18)(14,15,20,16)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=g,e^2=i,f^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,h^a=hi,c^b=ch,d^b=dh,e^b=ei,f^c=fi,h^c=hi,f^d=fi,h^d=hi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72271523, #72271558, #72271575 |
| maximal normal subgroup of | #72271523, #72271558, #72271575 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6352951);