Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
7568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(13,14,18,21)(15,16,23,19)(17,20,25,27)(22,24,28,26);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12);(6,9)(10,11)(13,15,18,23)(14,19,21,16)(17,22,25,28)(20,26,27,24);(7,12)(10,11)(13,15)(14,16)(17,22)(18,23)(19,21)(20,24)(25,28)(26,27);(5,7,8,12)(6,10,9,11)(13,16)(14,15)(17,24)(18,19)(20,22)(21,23)(25,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,17,18,25)(14,20,21,27)(15,22,23,28)(16,24,19,26)Polycyclic presentation:
2^9|a^2=gh,c^2=h,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72257999, #72281540, #72319710, … |
| maximal normal subgroup of | #72257999, #72281540, #72319710, … |
| maximal quotient of | #72282204 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7700578);