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 | |
7286 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(13,14,19,18)(15,20,23,28)(16,21,24,25)(17,26,27,22);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,23)(14,20,18,28)(16,17,24,27)(21,26,25,22);(6,8)(10,12)(14,18)(15,23)(16,24)(22,26);(7,11)(10,12)(13,16)(14,21)(15,17)(18,25)(19,24)(20,26)(22,28)(23,27);(5,7)(6,10)(8,12)(9,11)(13,17)(14,22)(15,24)(16,23)(18,26)(19,27)(20,21)(25,28);(5,8,9,6)(7,12,11,10)(13,18,19,14)(15,20,23,28)(16,25,24,21)(17,26,27,22)Polycyclic presentation:
2^9|a^2=h,b^2=h,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,f^c=fi,e^d=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72238242, #72238248, #72238253, … |
| maximal normal subgroup of | #72238242, #72238248, #72238253, … |
| maximal quotient of | #72256708, #72256746, #72264734 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7744351);