Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #1506685 |
| automorphism | ? |
| 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 | |
1840 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)(5,6)(7,9)(8,11)(10,12)(13,14,18,16)(15,19,17,20)(21,22,26,27)(23,28,29,35)(24,32,34,25)(30,31,33,36);(6,7)(8,12)(13,15,18,17)(14,19,16,20)(22,27)(23,29)(25,32)(30,33);(6,7)(10,11)(13,16,18,14)(15,19,17,20)(21,23)(22,28)(24,33)(25,31)(26,29)(27,35)(30,34)(32,36);(1,2,3,4)(5,7,9,6)(8,11,12,10)(13,17,18,15)(14,19,16,20)(21,24)(22,25)(23,30)(26,34)(27,32)(28,36)(29,33)(31,35);(5,8,9,12)(6,10,7,11)(21,25)(22,24)(23,31)(26,32)(27,34)(28,33)(29,36)(30,35)Polycyclic presentation:
2^9|a^2=h,b^2=fgh,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=dg,e^b=ef,d^c=df74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1554244);