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 | |
8329 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)(13,14)(15,23)(16,21)(17,18)(19,22)(20,27)(24,26)(25,28);(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,15,19,17)(14,18,22,23)(16,25,27,24)(20,28,21,26);(6,9)(10,11)(13,16,19,27)(14,20,22,21)(15,24,17,25)(18,26,23,28);(13,17,19,15)(14,18,22,23)(16,24,27,25)(20,28,21,26);(5,7)(6,10)(8,12)(9,11)(13,18)(14,15)(16,26)(17,22)(19,23)(20,24)(21,25)(27,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,21)(15,25)(17,24)(18,26)(19,27)(20,22)(23,28)Polycyclic presentation:
2^9|b^2=g,c^2=h,d^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,e^b=ei,d^c=dh,f^c=fi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72302966, #72302976, #72302981, … |
| maximal normal subgroup of | #72302966, #72302976, #72302981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7652041);