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 | |
7665 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)(7,10)(8,12)(13,14,18,22)(15,21,26,19)(16,20,27,17)(23,28,25,24);(1,2,3,4)(5,6)(7,12)(8,10)(9,11)(13,15)(14,19)(16,25)(17,24)(18,26)(20,28)(21,22)(23,27);(5,7)(6,8)(9,10)(11,12)(13,16)(14,20)(15,23)(17,22)(18,27)(19,24)(21,28)(25,26);(5,6)(7,12)(8,10)(9,11)(13,17,18,20)(14,16,22,27)(15,24,26,28)(19,25,21,23);(5,8,9,12)(6,10,11,7)(13,16)(14,20)(15,25)(17,22)(18,27)(19,28)(21,24)(23,26);(5,6)(7,8)(9,11)(10,12)(13,15)(14,21)(16,23)(17,24)(18,26)(19,22)(20,28)(25,27)Polycyclic presentation:
2^9|a^2=hi,b^2=ghi,d^2=hi,e^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72326930 |
| maximal normal subgroup of | #72326930 |
| maximal quotient of | #72327050, #72328328, #72329575, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7784155);