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 | |
7771 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)(13,14)(15,16)(17,21)(18,19)(20,22)(23,24)(25,28)(26,27);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12);(5,7)(6,9)(8,12)(10,11)(14,19)(15,22)(21,27)(23,28);(5,6)(7,10)(8,11)(9,12)(13,15)(14,20)(16,19)(17,23)(18,22)(21,25)(24,27)(26,28);(5,6)(7,9)(8,11)(10,12)(13,16,18,20)(14,15,19,22)(17,25,26,24)(21,28,27,23);(5,7)(6,10)(8,12)(9,11)(13,17)(14,21)(15,23)(16,24)(18,26)(19,27)(20,25)(22,28)Polycyclic presentation:
2^9|e^2=h,b^a=bg,c^a=ch,d^a=dh,c^b=ci,d^c=dhi,f^e=fhi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72326930 |
| maximal normal subgroup of | #72326930 |
| maximal quotient of | #72326938, #72328122, #72329469, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7566255);