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 | |
8479 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)(13,14)(15,20)(16,18)(17,19);(5,6,10,9)(7,12,8,11)(13,15,17,16)(14,18,19,20);(2,4)(5,7,10,8)(6,11,9,12)(13,14)(15,20)(16,18)(17,19);(1,2)(3,4);(5,8,10,7)(6,11,9,12)(13,14)(15,18)(16,20)(17,19);(5,9,10,6)(7,12,8,11)(13,16,17,15)(14,18,19,20)Polycyclic presentation:
2^9|b^2=i,c^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 | #72238159, #72238163, #72238164, … |
| maximal normal subgroup of | #72238159, #72238163, #72238164, … |
| maximal quotient of | #72265081, C2×#7684682, #72281102, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7591878);