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 | |
7692 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(13,14)(15,21)(16,20)(17,18)(19,23)(22,27)(24,25)(26,28);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12);(6,9)(10,11)(14,18)(19,21)(20,25)(26,27);(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,28)(25,27);(5,7)(6,10)(8,12)(9,11)(13,16,17,24)(14,20,18,25)(15,22,23,28)(19,26,21,27)Polycyclic presentation:
2^9|a^2=g,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72326930 |
| maximal normal subgroup of | #72326930 |
| maximal quotient of | #72327051, #72328335, #72329578, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7629890);