Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7200022 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 ◁ #2392 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1532 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,15)(11,14,12,16)(17,18)(19,28)(20,25)(21,26)(22,24)(23,30)(27,35)(29,40)(31,38)(32,37)(33,41)(34,39)(36,44)(42,46)(43,48)(45,47);(1,3)(2,5)(4,7)(6,8)(9,10)(11,16)(12,14)(13,15)(17,19)(18,23)(20,32)(21,30)(22,31)(24,36)(25,40)(26,28)(27,38)(29,33)(34,45)(35,44)(37,41)(39,48)(42,47)(43,46);(3,7)(5,8)(9,11,13,12)(10,14,15,16)(18,24)(22,35)(23,36)(25,39)(26,27)(28,38)(31,44)(34,46)(37,47)(40,48)(41,42)(43,45);(2,6)(3,7)(9,12,13,11)(10,14,15,16)(17,20)(18,25)(19,29)(21,33)(22,34)(23,37)(24,39)(26,41)(27,42)(28,40)(30,32)(31,43)(35,46)(36,47)(38,48)(44,45);(3,7)(5,8)(10,15)(14,16)(18,26)(19,30)(20,33)(24,27)(31,44)(34,46)(37,40)(47,48)Polycyclic presentation:
2^9|a^2=fi,c^2=fi,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,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, 7111424);