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 | |
9964 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 20 points:
(2,4)(6,10)(8,11)(13,14)(15,17)(16,18)(19,20);(1,2)(3,4)(5,6)(7,11)(8,12)(9,10)(13,15)(14,17)(16,20)(18,19);(6,10)(7,12)(14,16)(15,19);(13,15)(14,17)(16,20)(18,19);(5,7,9,12)(6,11,10,8)(13,16,18,14)(15,20,19,17);(5,8,9,11)(6,12,10,7)(13,17,18,20)(14,19,16,15)Polycyclic presentation:
2^9|e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72232031, #72233520, #72233694, … |
| maximal normal subgroup of | #72232031, #72233520, #72233694, … |
| maximal quotient of | #72232049, #72232079, #72232269, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7619845);