Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #917169 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,3)(6,7)(8,12)(13,14)(15,20)(16,17)(18,19)(22,25)(26,28);(2,3)(5,6,9,7)(8,10,12,11)(13,15)(14,17)(16,18)(19,20)(21,22)(23,26)(24,25)(27,28);(1,2,4,3)(6,7)(10,11)(14,19)(15,16)(23,27)(26,28);(1,3,4,2)(5,7,9,6)(8,11,12,10)(13,16,18,15)(14,17,19,20)(21,23)(22,26)(24,27)(25,28);(5,8)(6,10)(7,11)(9,12)(13,17)(14,15)(16,19)(18,20)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^9|b^2=gh,c^2=ghi,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg,d^c=df80 conjugacy classes — view the full character table.
| maximal subgroup of | #72300470, #72300502, #72300539, … |
| maximal normal subgroup of | #72300470, #72300502, #72300539, … |
| maximal quotient of | #72330827, #72330956 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 838853);