Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 36 points:
(2,4)(6,10)(7,11)(13,14,18,16)(15,19,17,20)(21,22,26,27)(23,25,30,28)(24,32,34,29)(31,33,36,35);(5,6)(7,12)(8,11)(9,10)(13,15,18,17)(14,19,16,20)(22,27)(23,30)(29,32)(31,36);(6,10)(8,12)(13,16,18,14)(15,19,17,20)(21,23,26,30)(22,28,27,25)(24,31,34,36)(29,33,32,35);(1,2,3,4)(5,7)(6,8)(9,11)(10,12)(13,17,18,15)(14,19,16,20)(21,24,26,34)(22,29,27,32)(23,31,30,36)(25,35,28,33);(5,8)(6,11)(7,10)(9,12)(21,25,26,28)(22,23,27,30)(24,33,34,35)(29,36,32,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=fh,c^2=fgh,d^2=fgi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3375192);