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 | |
8460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(13,14,19,16)(15,22,21,18)(17,24,26,20)(23,27,25,28);(1,2)(3,4)(13,15)(14,18)(16,22)(17,25)(19,21)(20,28)(23,26)(24,27);(6,10)(7,11)(14,16)(15,21)(17,26)(27,28);(5,7,9,11)(6,8,10,12)(13,16,19,14)(15,22,21,18)(17,24,26,20)(23,28,25,27);(5,8)(6,11)(7,10)(9,12)(13,17)(14,20)(15,23)(16,24)(18,27)(19,26)(21,25)(22,28);(13,18,19,22)(14,21,16,15)(17,27,26,28)(20,25,24,23)Polycyclic presentation:
2^9|a^2=i,d^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72238223, #72238237, #72238245, … |
| maximal normal subgroup of | #72238223, #72238237, #72238245, … |
| maximal quotient of | #72257920 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7627556);