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 | |
8488 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,3,4)(5,6,8,9)(7,10,12,11)(13,14,18,17)(15,20,16,19);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(14,17)(19,20);(6,9)(10,11)(13,14,18,17)(15,19,16,20);(5,6)(7,10)(8,9)(11,12)(13,15,18,16)(14,19,17,20);(5,6)(7,11)(8,9)(10,12)(13,16,18,15)(14,19,17,20);(5,7,8,12)(6,10,9,11)(13,17,18,14)(15,20,16,19)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=hi,e^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 | #72297477, C2×#7989633, #72308960, … |
| maximal normal subgroup of | #72297477, C2×#7989633, #72308960, … |
| maximal quotient of | #72297542, #72297544, #72297596, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7896829);