Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
3164 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)(7,12)(8,10)(9,11)(13,14,16,17)(15,19,20,18)(21,22)(23,24)(25,28)(26,27);(5,7,10,9)(6,11,8,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,25)(24,27)(26,28);(7,9)(11,12)(14,17)(18,19)(21,24,26,25)(22,23,27,28);(1,2,3,4)(5,8,10,6)(7,11,9,12)(13,15)(14,18)(16,20)(17,19)(21,25,26,24)(22,23,27,28);(5,9,10,7)(6,11,8,12)(13,14)(15,18)(16,17)(19,20)(21,23)(22,24)(25,27)(26,28)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=fh,d^2=gi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72328920, #72328935, #72328975, … |
| maximal normal subgroup of | #72328920, #72328935, #72328975, … |
| maximal quotient of | #72329925, #72330087, #72332816 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2227901);