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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2388 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2130 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14,19,24)(15,20,25,35)(16,22,30,38)(17,23,18,21)(26,33,39,43)(27,32,28,34)(29,36,31,37)(40,44,41,42);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,20)(16,26)(17,27)(18,28)(19,25)(21,34)(22,33)(23,32)(24,35)(29,40)(30,39)(31,41)(36,44)(37,42)(38,43);(5,7,8,12)(6,9,10,11)(14,21)(15,25)(17,18)(20,32)(22,36)(23,24)(26,39)(29,31)(33,42)(34,35)(37,38)(43,44);(2,4)(7,12)(9,11)(13,16)(14,22)(15,26)(17,31)(18,29)(19,30)(20,33)(21,36)(23,37)(24,38)(25,39)(27,41)(28,40)(32,42)(34,44)(35,43);(6,10)(9,11)(13,17,19,18)(14,21,24,23)(15,27,25,28)(16,29,30,31)(20,34,35,32)(22,37,38,36)(26,40,39,41)(33,42,43,44)Polycyclic presentation:
2^9|a^2=i,c^2=hi,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,e^c=ei,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, 6961580);