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×#525 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3185 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)(13,14,19,27,18,23,16,21)(15,24,26,20,17,22,25,28);(5,6,8,9)(7,10,11,12)(13,15,18,17)(14,20,23,28)(16,26,19,25)(21,22,27,24);(2,4)(6,9)(7,11)(13,16,18,19)(14,21,23,27)(15,25,17,26)(20,24,28,22);(5,7)(6,10)(8,11)(9,12)(13,17,18,15)(14,22,23,24)(16,26,19,25)(20,27,28,21);(6,9)(7,11)(14,23)(15,17)(21,27)(25,26)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=i,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh77 conjugacy classes — view the full character table.
| maximal subgroup of | #72263948, #72263950, #72264047 |
| maximal normal subgroup of | #72263948, #72263950, #72264047 |
| maximal quotient of | #72307946 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6503598);