Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3338 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2)(3,4)(5,6,8,9)(7,11,12,10)(13,14)(15,16)(17,20)(18,19)(21,22)(23,28)(24,26)(25,27);(1,2,3,4)(6,9)(7,12)(13,15,17,19)(14,18,20,16)(21,23)(22,24)(25,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,28)(24,26)(25,27);(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,15)(14,16)(17,19)(18,20)(21,22)(23,24)(25,27)(26,28);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,16)(14,19)(15,20)(17,18)(21,24)(22,26)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=fhi,b^2=gi,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232893, #72232901, #72232915, … |
| maximal normal subgroup of | #72232893, #72232901, #72232915, … |
| maximal quotient of | #72328599, #72328605, #72329972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2751346);