Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4033837 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
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 | |
4255 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)(5,6,9,12)(7,8,11,10)(14,17)(15,19)(22,25)(23,27);(5,7)(6,10)(8,12)(9,11)(15,19)(18,20)(21,22,24,25)(23,28,27,26);(7,11)(8,10)(13,14)(15,18)(16,17)(19,20)(21,22,24,25)(23,26,27,28);(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)(13,14)(15,18)(16,17)(19,20)(21,23,24,27)(22,26,25,28);(5,8)(6,11)(7,12)(9,10)(13,15)(14,18)(16,19)(17,20)(21,22)(23,28)(24,25)(26,27)Polycyclic presentation:
2^9|a^2=h,b^2=fh,c^2=fh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef80 conjugacy classes — view the full character table.
| maximal subgroup of | #72232481, #72232510, #72232514, … |
| maximal normal subgroup of | #72232481, #72232510, #72232514, … |
| maximal quotient of | #72328658, #72329930, #72331307, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3951905);