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 | |
7708 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)(6,10)(7,11)(13,14,18,20)(15,21,23,17)(16,19,25,28)(22,24,27,26);(1,2)(3,4)(5,6)(7,8)(9,10)(11,12)(13,15)(14,17)(16,22)(18,23)(19,26)(20,21)(24,28)(25,27);(6,10)(7,11)(13,15,18,23)(14,17,20,21)(16,22,25,27)(19,26,28,24);(7,11)(8,12)(13,16)(14,19)(15,22)(17,26)(18,25)(20,28)(21,24)(23,27);(5,7)(6,8)(9,11)(10,12)(13,15)(14,17)(16,22)(18,23)(19,26)(20,21)(24,28)(25,27);(5,8)(6,7)(9,12)(10,11)(13,17)(14,15)(16,24)(18,21)(19,27)(20,23)(22,28)(25,26)Polycyclic presentation:
2^9|a^2=h,c^2=h,b^a=bg,c^a=ch,e^a=ei,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 | #72327640 |
| maximal normal subgroup of | #72327640 |
| maximal quotient of | #72327784, #72328008, #72328229, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7977019);