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 | |
7758 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)(5,6)(7,8)(9,10)(11,12)(13,14,19,21)(15,22,16,18)(17,20,26,28)(23,27,24,25);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12);(7,11)(8,12)(13,15,19,16)(14,18,21,22)(17,23,26,24)(20,25,28,27);(6,10)(8,12)(13,16,19,15)(14,18,21,22)(17,23,26,24)(20,27,28,25);(5,8)(6,7)(9,12)(10,11)(13,17)(14,20)(15,23)(16,24)(18,25)(19,26)(21,28)(22,27);(5,6)(7,8)(9,10)(11,12)(13,18)(14,15)(16,21)(17,25)(19,22)(20,23)(24,28)(26,27)Polycyclic presentation:
2^9|a^2=gh,c^2=h,d^2=h,b^a=bg,c^a=ch,d^a=di,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi95 conjugacy classes — view the full character table.
| maximal subgroup of | #72282562, #72312523, #72312537, … |
| maximal normal subgroup of | #72282562, #72312523, #72312537, … |
| maximal quotient of | #72323321, #72323397, C2×#8063108, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7970304);