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 | |
8695 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)(5,6,9,11)(7,8,12,10)(13,14,18,22)(15,19,24,28)(16,25,26,20)(17,21,23,27);(1,2)(3,4)(13,15,18,24)(14,19,22,28)(16,23,26,17)(20,21,25,27);(5,7,9,12)(6,10,11,8)(13,16,18,26)(14,20,22,25)(15,17,24,23)(19,27,28,21);(5,7)(6,8)(9,12)(10,11)(13,16)(14,20)(15,23)(17,24)(18,26)(19,21)(22,25)(27,28);(5,8)(6,7)(9,10)(11,12)(13,14)(15,19)(16,20)(17,27)(18,22)(21,23)(24,28)(25,26);(13,17,18,23)(14,21,22,27)(15,16,24,26)(19,25,28,20)Polycyclic presentation:
2^9|a^2=h,b^2=i,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 | #72238149, #72238202, #72238241, … |
| maximal normal subgroup of | #72238149, #72238202, #72238241, … |
| maximal quotient of | #72297183, #72297519, #72297613, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7846337);