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 | |
7927 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,4,3)(5,6,10,8)(7,12,9,11)(14,15)(18,19);(5,7,10,9)(6,11,8,12)(13,14,17,15)(16,18,20,19);(2,3)(5,6,10,8)(7,12,9,11)(13,15,17,14)(16,18,20,19);(1,3,4,2)(13,16,17,20)(14,18,15,19);(5,8,10,6)(7,12,9,11);(5,9,10,7)(6,11,8,12)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=h,e^2=i,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi113 conjugacy classes — view the full character table.
| maximal subgroup of | #72315027, #72315196, #72315206 |
| maximal normal subgroup of | #72315027, #72315196, #72315206 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7595106);