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 | |
8810 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)(7,11)(10,12)(15,22)(17,25)(19,20)(21,24);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14,18,16)(15,20,22,19)(17,21,25,24)(23,27,28,26);(6,9)(10,12)(13,15,18,22)(14,19,16,20)(17,23,25,28)(21,26,24,27);(13,16,18,14)(15,19,22,20)(17,24,25,21)(23,26,28,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,20)(16,19)(17,23)(18,22)(21,27)(24,26)(25,28);(5,7)(6,10)(8,11)(9,12)(13,17)(14,21)(15,23)(16,24)(18,25)(19,26)(20,27)(22,28)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei104 conjugacy classes — view the full character table.
| maximal subgroup of | #72304554, #72304562, #72304569, … |
| maximal normal subgroup of | #72304554, #72304562, #72304569, … |
| maximal quotient of | #72336602, #72336614 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7615716);