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 | |
8595 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)(9,12)(10,11)(13,14,18,20)(15,23,24,19)(16,27,26,21)(17,22,25,28);(1,2)(3,4)(6,11)(7,9)(13,15,18,24)(14,19,20,23)(16,25,26,17)(21,28,27,22);(5,7,10,9)(6,8,11,12)(14,20)(15,24)(16,26)(22,28);(5,8)(6,9)(7,11)(10,12)(13,14)(15,19)(16,21)(17,22)(18,20)(23,24)(25,28)(26,27);(5,8)(6,7)(9,11)(10,12)(13,16)(14,21)(15,25)(17,24)(18,26)(19,28)(20,27)(22,23);(5,9,10,7)(6,12,11,8)(13,17,18,25)(14,22,20,28)(15,26,24,16)(19,27,23,21)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72238145, #72238204, #72238241, … |
| maximal normal subgroup of | #72238145, #72238204, #72238241, … |
| maximal quotient of | #72262654, #72263594 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7892318);