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 | |
8913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6,9,11)(7,8,12,10)(13,14,17,18)(15,21,23,19)(16,20,25,26)(22,28,24,27);(1,2)(3,4)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,28)(25,26);(5,7,9,12)(6,10,11,8)(14,18)(19,21)(20,26)(27,28);(5,7)(6,8)(9,12)(10,11)(13,14)(15,19)(16,20)(17,18)(21,23)(22,27)(24,28)(25,26);(5,8)(6,7)(9,10)(11,12)(13,15)(14,19)(16,24)(17,23)(18,21)(20,28)(22,25)(26,27);(13,16,17,25)(14,20,18,26)(15,22,23,24)(19,27,21,28)Polycyclic presentation:
2^9|a^2=h,c^2=hi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72238167, #72238199, #72238234, … |
| maximal normal subgroup of | #72238167, #72238199, #72238234, … |
| maximal quotient of | C2×#7905733, #72327006, #72327970, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7812929);