Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996639 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2552 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)(7,11)(10,12)(13,14,18,16,19,22,17,20)(15,21,25,24,23,28,26,27);(6,9)(7,11)(14,20)(15,23)(16,22)(17,18)(21,24)(27,28);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,23)(14,21,22,28)(16,24,20,27)(17,26,18,25);(6,9)(10,12)(13,16,17,14,19,20,18,22)(15,24,26,21,23,27,25,28);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,16,22,20)(15,25,23,26)(21,27,28,24)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=fi,e^2=h,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72283930, #72284255, #72284702, … |
| maximal normal subgroup of | #72283930, #72284255, #72284702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7053329);