Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6682596 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1935 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6,10,14)(7,15,17,11)(8,12,9,13)(16,20,18,19)(21,22)(23,30)(24,34)(25,29)(26,28)(27,32)(31,36)(33,35);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23,26,32)(22,27,28,30)(24,35,29,31)(25,33,34,36);(2,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,34,22,26,29,25,28)(23,31,36,30,32,35,33,27);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(22,28)(23,32)(24,29)(33,36);(2,4)(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,25,26,34)(22,29,28,24)(23,33,32,36)(27,31,30,35)Polycyclic presentation:
2^9|a^2=fi,b^2=i,c^2=fg,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^c=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307339, #72307685, #72307780, … |
| maximal normal subgroup of | #72307339, #72307685, #72307780, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6594961);