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 | C2×#3475 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3243 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,10)(8,9)(11,12)(14,19)(16,17)(20,23)(21,22)(24,25)(27,28);(1,2,3,4)(13,14,16,19,18,21,17,22)(15,23,24,20,26,28,25,27);(6,9)(10,12)(13,15,18,26)(14,20,21,27)(16,25,17,24)(19,23,22,28);(14,19)(16,17)(20,23)(21,22)(24,25)(27,28);(5,7,8,11)(6,10,9,12)(14,21)(19,22)(20,27)(23,28)Polycyclic presentation:
2^9|b^2=f,c^2=i,e^2=g,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 | #72263899, #72263913, #72264047 |
| maximal normal subgroup of | #72263899, #72263913, #72264047 |
| maximal quotient of | #72264477, #72264935 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6361927);