Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6993947 |
| 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 |
| prime radical | #23871 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2395 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2099 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2)(3,4)(5,6)(7,9)(8,10)(11,12)(13,14,18,23)(15,24,26,20)(16,21,30,37)(17,19,31,22)(25,38,29,35)(27,36,41,33)(28,39,32,34)(40,43,42,44);(5,6)(7,11)(8,10)(9,12)(14,19)(16,28)(17,31)(20,33)(21,37)(22,23)(24,36)(25,40)(27,41)(29,42)(30,32)(35,38);(2,4)(5,7,8,12)(6,9,10,11)(13,15,18,26)(14,20,23,24)(16,29,30,25)(17,27,31,41)(19,33,22,36)(21,38,37,35)(28,42,32,40)(34,44,39,43);(7,12)(9,11)(13,16,18,30)(14,21,23,37)(15,25,26,29)(17,32,31,28)(19,34,22,39)(20,35,24,38)(27,42,41,40)(33,43,36,44);(6,10)(9,11)(15,26)(20,24)(25,29)(27,41)(33,36)(35,38)(40,42)(43,44)Polycyclic presentation:
2^9|a^2=i,c^2=hi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^d=fi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6961773);