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 | |
8612 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)(5,6,9,10)(7,11,12,8)(13,14,19,23)(15,20,18,22)(16,17,27,21)(24,25,26,28);(1,2)(3,4)(6,10)(7,12)(13,15,19,18)(14,20,23,22)(16,24,27,26)(17,25,21,28);(5,7)(6,11)(8,10)(9,12)(13,16,19,27)(14,21,23,17)(15,24,18,26)(20,28,22,25);(5,8)(6,12)(7,10)(9,11)(13,16)(14,17)(15,24)(18,26)(19,27)(20,25)(21,23)(22,28);(5,8)(6,7)(9,11)(10,12)(13,17)(14,16)(15,25)(18,28)(19,21)(20,24)(22,26)(23,27);(5,7)(6,11)(8,10)(9,12)(13,18,19,15)(14,22,23,20)(16,26,27,24)(17,28,21,25)Polycyclic presentation:
2^9|a^2=hi,b^2=h,c^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi101 conjugacy classes — view the full character table.
| maximal subgroup of | #72238149, #72238218, #72238235, … |
| maximal normal subgroup of | #72238149, #72238218, #72238235, … |
| maximal quotient of | #72297191, #72297569, #72297617, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7875969);