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 | |
8171 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 20 points:
(1,2,4,3)(5,6)(7,8)(9,12)(10,11)(14,15)(18,19);(5,7,9,11)(6,10,12,8)(13,14,17,15)(16,18,20,19);(2,3)(5,8)(6,11)(7,12)(9,10)(13,15,17,14)(16,18,20,19);(1,3,4,2)(5,7,9,11)(6,10,12,8)(13,16,17,20)(14,18,15,19);(5,8)(6,7)(9,10)(11,12);(5,6)(7,10)(8,11)(9,12)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=gi,d^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^c=dh,f^e=fi107 conjugacy classes — view the full character table.
| maximal subgroup of | #72234856, #72234858, #72234891, … |
| maximal normal subgroup of | #72234856, #72234858, #72234891, … |
| maximal quotient of | #72262088, #72265075, C2×#7697273 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7604469);