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 | |
8390 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)(7,8)(9,10)(11,12);(1,2)(3,4)(14,19)(15,16)(17,25)(27,28);(6,10)(7,11)(13,14)(15,20)(16,21)(17,26)(18,19)(22,25)(23,28)(24,27);(5,7,9,11)(6,8,10,12)(13,15,18,16)(14,20,19,21)(17,23,25,24)(22,27,26,28);(5,8)(6,11)(7,10)(9,12)(13,16,18,15)(14,21,19,20)(17,24,25,23)(22,28,26,27);(13,17)(14,22)(15,23)(16,24)(18,25)(19,26)(20,27)(21,28)Polycyclic presentation:
2^9|d^2=hi,e^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72281935, #72281978, #72282768, … |
| maximal normal subgroup of | #72281935, #72281978, #72282768, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7713497);