Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6573615 |
| 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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3219 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(13,14,16,20,18,21,17,22)(15,23,25,28,24,27,26,19);(2,4)(5,6,8,9)(7,10,11,12)(13,15,18,24)(14,19,21,28)(16,26,17,25)(20,27,22,23);(6,9)(7,11)(14,20)(16,17)(19,27)(21,22)(23,28)(25,26);(5,7)(6,10)(8,11)(9,12)(14,21)(15,24)(20,22)(25,26);(6,9)(7,11)(14,21)(19,28)(20,22)(23,27)Polycyclic presentation:
2^9|a^2=f,b^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72257482, #72257494, #72257553, … |
| maximal normal subgroup of | #72257482, #72257494, #72257553, … |
| maximal quotient of | #72264740, #72265107 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503744);