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 | |
1955 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(5,6)(7,15)(8,12)(9,13)(10,14)(11,17)(16,20)(18,19)(21,22)(23,30)(24,25)(26,29)(27,36)(28,32)(31,33)(34,35);(1,2)(3,4)(5,7)(6,11)(8,16)(9,18)(10,17)(12,19)(13,20)(14,15)(21,23)(22,28)(24,35)(25,33)(26,32)(27,34)(29,30)(31,36);(2,4)(5,8,10,9)(6,12,14,13)(7,16,17,18)(11,19,15,20)(21,24,25,22,26,36,27,29)(23,31,33,30,32,35,34,28);(5,8,10,9)(6,13,14,12)(7,16,17,18)(11,20,15,19)(22,29)(23,32)(24,36)(33,34);(2,4)(5,9,10,8)(6,13,14,12)(7,16,17,18)(11,19,15,20)(21,25,26,27)(22,24,29,36)(23,33,32,34)(28,35,30,31)Polycyclic presentation:
2^9|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 | #72307435, #72307490, #72307888, … |
| maximal normal subgroup of | #72307435, #72307490, #72307888, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6594556);