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 | |
7844 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(13,14)(15,18)(16,23)(17,19)(20,25)(21,27)(22,28)(24,26);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15,17,21)(14,18,19,27)(16,24,25,22)(20,28,23,26);(6,9)(10,11)(14,19)(15,21)(20,23)(22,24);(5,7)(6,10)(8,12)(9,11)(13,15)(14,18)(16,22)(17,21)(19,27)(20,26)(23,28)(24,25);(5,7)(6,10)(8,12)(9,11)(13,16)(14,20)(15,22)(17,25)(18,26)(19,23)(21,24)(27,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,20)(17,19)(21,27)(22,26)(23,25)(24,28)Polycyclic presentation:
2^9|a^2=g,b^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi98 conjugacy classes — view the full character table.
| maximal subgroup of | #72326380 |
| maximal normal subgroup of | #72326380 |
| maximal quotient of | #72326391, #72326415, #72327164, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7659350);