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 | |
8583 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)(15,20)(16,21)(17,19)(18,22)(23,27)(24,28)(25,26);(1,2)(3,4)(5,6)(7,11)(8,9)(10,12)(13,15,19,16)(14,20,17,21)(18,24,26,23)(22,28,25,27);(6,9)(10,11)(14,17)(15,16)(22,25)(23,24);(13,16,19,15)(14,21,17,20)(18,24,26,23)(22,28,25,27);(5,6)(7,10)(8,9)(11,12)(13,17,19,14)(15,21,16,20)(18,25,26,22)(23,28,24,27);(5,7)(6,10)(8,12)(9,11)(13,18,19,26)(14,22,17,25)(15,23,16,24)(20,27,21,28)Polycyclic presentation:
2^9|b^2=h,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72297863, C2×#7722991 |
| maximal normal subgroup of | #72297863, C2×#7722991 |
| maximal quotient of | #72297878, #72297963, #72298083, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7630187);