Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212×C4 |
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 | |
10316 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)(7,9)(8,10)(11,12);(1,2)(3,4)(5,6,9,7)(8,11,12,10)(13,14,17,15)(16,19,20,18);(6,7)(10,11)(14,15)(18,19);(13,14,17,15)(16,18,20,19);(5,7,9,6)(8,11,12,10)(13,15,17,14)(16,19,20,18);(5,8,9,12)(6,10,7,11)(13,16,17,20)(14,18,15,19)Polycyclic presentation:
2^9|a^2=gh,b^2=i,d^2=hi,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72233749, #72233765, #72234448, … |
| maximal normal subgroup of | #72233749, #72233765, #72234448, … |
| maximal quotient of | #72261932, #72262210, #72297244, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7646585);