Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6994139 |
| 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 | |
1427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,6)(7,8)(9,10,14,11)(12,15,13,16)(17,18)(19,24)(20,26)(21,27)(22,25)(23,28)(29,32)(30,31);(1,2)(3,8)(4,7)(5,6)(10,11)(12,13)(17,19,23,20)(18,24,28,26)(21,30,22,29)(25,32,27,31);(9,11,14,10)(12,15,13,16)(18,25)(19,20)(21,22)(24,31)(26,32)(27,28);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,15,11,16)(17,20,23,19)(18,26,28,24)(21,29,22,30)(25,31,27,32);(1,4,5,3)(2,7,6,8)(9,13,14,12)(10,15,11,16)(17,21,23,22)(18,25,28,27)(19,29,20,30)(24,31,26,32)Polycyclic presentation:
2^9|a^2=hi,b^2=i,c^2=hi,d^2=fi,e^2=fi,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, 7015278);