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 | |
8606 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 28 points:
(2,4)(5,6)(7,11)(8,10)(9,12)(13,14,18,17)(15,19,21,23)(16,20,24,25)(22,26,27,28);(1,2)(3,4)(13,15,18,21)(14,19,17,23)(16,22,24,27)(20,26,25,28);(5,6,8,10)(7,9,12,11)(15,21)(19,23)(22,27)(26,28);(5,6)(7,9)(8,10)(11,12)(13,16)(14,20)(15,22)(17,25)(18,24)(19,26)(21,27)(23,28);(5,7)(6,9)(8,12)(10,11)(13,17,18,14)(15,23,21,19)(16,20,24,25)(22,26,27,28);(13,15)(14,19)(16,22)(17,23)(18,21)(20,26)(24,27)(25,28)Polycyclic presentation:
2^9|a^2=i,b^2=i,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, #72256676, #72261985, … |
| maximal normal subgroup of | #72256602, #72256676, #72261985, … |
| maximal quotient of | #72256752, #72263448, #72282572 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7738005);