Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | |
2178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 44 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 44 points:
(2,4)(5,6)(7,9)(8,10)(11,12)(13,14)(15,20)(16,22)(17,23)(18,21)(19,24)(25,33)(26,34)(27,32)(28,35)(29,36)(30,38)(31,37)(39,42)(40,44)(41,43);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,15,19,28)(14,20,24,35)(16,25,30,40)(17,26,18,27)(21,32,23,34)(22,33,38,44)(29,39,31,41)(36,42,37,43);(5,7,8,12)(6,9,10,11)(14,21)(17,18)(20,32)(22,36)(23,24)(26,27)(29,31)(33,42)(34,35)(37,38)(39,41)(43,44);(2,4)(7,12)(9,11)(13,16)(14,22)(15,25)(17,31)(18,29)(19,30)(20,33)(21,36)(23,37)(24,38)(26,41)(27,39)(28,40)(32,42)(34,43)(35,44);(6,10)(9,11)(13,17,19,18)(14,23,24,21)(15,26,28,27)(16,29,30,31)(20,34,35,32)(22,36,38,37)(25,39,40,41)(33,42,44,43)Polycyclic presentation:
2^9|b^2=i,c^2=h,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=eh,d^c=dh,e^c=ei,g^c=gi,e^d=ei71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6961184);