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 | #45241 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2025 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(1,2,4,3)(5,6)(7,9)(8,10)(11,12)(13,14,19,25)(15,26,29,21)(16,30,17,23)(18,24,20,35)(22,33,40,31)(27,41,32,37)(28,39,34,38)(36,44,42,43);(2,3)(5,6)(7,11)(8,10)(9,12)(13,15,19,29)(14,21,25,26)(16,27,17,32)(18,34,20,28)(22,36,40,42)(23,41,30,37)(24,39,35,38)(31,44,33,43);(5,7,8,12)(6,9,10,11)(13,16)(14,22)(15,27)(17,19)(18,31)(20,33)(21,36)(23,24)(25,40)(26,42)(28,43)(29,32)(30,35)(34,44)(37,38)(39,41);(1,3,4,2)(7,12)(9,11)(13,17,19,16)(14,23,25,30)(15,27,29,32)(18,31,20,33)(21,37,26,41)(22,35,40,24)(28,44,34,43)(36,39,42,38);(6,10)(9,11)(13,18,19,20)(14,24,25,35)(15,28,29,34)(16,31,17,33)(21,38,26,39)(22,23,40,30)(27,43,32,44)(36,37,42,41)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6963300);