Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6,8,9)(7,10,11,12)(13,14,16,18)(15,19,20,17)(21,22,25,24)(23,26,28,27);(6,9)(7,11)(13,14,16,18)(15,17,20,19)(21,22)(23,26)(24,25)(27,28);(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,23,25,28)(22,26,24,27);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,24,25,22)(23,27,28,26);(5,6)(7,12)(8,9)(10,11)(13,15)(14,17)(16,20)(18,19)(21,23)(22,27)(24,26)(25,28)Polycyclic presentation:
2^9|a^2=fg,b^2=g,c^2=fgh,d^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
| maximal subgroup of | #72329290, #72329312, #72329394, … |
| maximal normal subgroup of | #72329290, #72329312, #72329394, … |
| maximal quotient of | #72330419, #72333268 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3775563);