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 | |
8114 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12);(1,2)(3,4)(13,14)(15,20)(16,17)(18,19)(21,26)(22,28)(23,25)(24,27);(5,6,8,10)(7,9,12,11)(14,19)(17,26)(20,27)(23,28);(5,6)(7,9)(8,10)(11,12)(13,15)(14,20)(16,25)(17,23)(18,24)(19,27)(21,22)(26,28);(5,7)(6,9)(8,12)(10,11)(13,16,18,21)(14,17,19,26)(15,22,24,25)(20,28,27,23);(13,17)(14,21)(15,23)(16,19)(18,26)(20,22)(24,28)(25,27)Polycyclic presentation:
2^9|a^2=g,c^2=h,e^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 | #72256602, #72256674, #72282531 |
| maximal normal subgroup of | #72256602, #72256674, #72282531 |
| maximal quotient of | #72256746, #72263446, #72283061 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7733706);