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 | |
8991 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)(14,18)(15,21)(20,26)(22,23);(1,2)(3,4)(13,14)(15,19)(16,20)(17,18)(21,25)(22,27)(23,28)(24,26);(5,7,9,12)(6,10,11,8)(13,14,17,18)(15,19,21,25)(16,20,24,26)(22,27,23,28);(5,7)(6,8)(9,12)(10,11)(13,14)(15,19)(16,20)(17,18)(21,25)(22,27)(23,28)(24,26);(5,8)(6,7)(9,10)(11,12)(13,15)(14,19)(16,23)(17,21)(18,25)(20,28)(22,24)(26,27);(13,16,17,24)(14,20,18,26)(15,22,21,23)(19,27,25,28)Polycyclic presentation:
2^9|a^2=hi,c^2=h,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 | #72236842, #72236863, #72238145, … |
| maximal normal subgroup of | #72236842, #72236863, #72238145, … |
| maximal quotient of | #72263600, #72263606, C2×#7918405, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7825601);