Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
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 ◁ C2×#532 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,19)(16,20)(21,22,27,32)(23,33,24,30)(25,28,26,31)(29,35,36,34);(2,4)(5,7)(6,10)(8,15)(9,14)(11,19)(12,18)(13,16)(17,20)(22,28)(23,24)(25,26)(29,33)(30,36)(31,32);(5,8,9,16)(6,11,12,20)(7,13,14,15)(10,17,18,19)(21,23,27,24)(22,29,32,36)(25,34,26,35)(28,30,31,33);(7,14)(10,18)(13,15)(17,19)(21,24,27,23)(22,30,32,33)(25,34,26,35)(28,29,31,36);(6,12)(10,18)(11,20)(17,19)(21,25,27,26)(22,31,32,28)(23,34,24,35)(29,33,36,30)Polycyclic presentation:
2^9|a^2=i,c^2=h,d^2=i,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^b=ei,f^b=fi,g^b=gi,f^c=fi,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
| maximal subgroup of | #72250301, #72250923, #72257082, … |
| maximal normal subgroup of | #72250301, #72250923, #72257082, … |
| maximal quotient of | #72264747, #72307910 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6384773);