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 | |
8692 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)(7,8)(9,10)(11,12)(13,14)(15,22)(16,20)(17,19)(18,25)(21,26)(23,27)(24,28);(1,2)(3,4)(13,15,19,16)(14,20,17,22)(18,23,26,24)(21,27,25,28);(6,10)(7,11)(14,17)(15,16)(21,25)(23,24);(5,7,9,11)(6,8,10,12)(13,16,19,15)(14,20,17,22)(18,24,26,23)(21,27,25,28);(5,8)(6,11)(7,10)(9,12)(13,17,19,14)(15,20,16,22)(18,25,26,21)(23,27,24,28);(13,18)(14,21)(15,23)(16,24)(17,25)(19,26)(20,27)(22,28)Polycyclic presentation:
2^9|b^2=i,d^2=h,e^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei98 conjugacy classes — view the full character table.
| maximal subgroup of | #72257179, #72257669, #72312182, … |
| maximal normal subgroup of | #72257179, #72257669, #72312182, … |
| maximal quotient of | #72257927 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7625604);