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 | |
7918 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)(7,12)(8,10)(9,11)(14,18)(19,20);(5,7,9,8)(6,10,11,12)(13,14,17,18)(15,19,16,20);(2,4)(5,6)(7,12)(8,10)(9,11);(1,2)(3,4)(13,15,17,16)(14,19,18,20);(5,6)(7,10)(8,12)(9,11)(13,16,17,15)(14,20,18,19);(5,8,9,7)(6,10,11,12)Polycyclic presentation:
2^9|b^2=g,d^2=gi,e^2=gi,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 | #72236865, #72238167, #72238201, … |
| maximal normal subgroup of | #72236865, #72238167, #72238201, … |
| maximal quotient of | C2×#7684936, #72281575, #72281642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7592132);