Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6437418 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3045 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2551 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,12)(8,10)(9,11)(13,14,17,21)(15,19,22,26)(16,18,24,20)(23,25,28,27);(7,8)(10,12)(14,18)(16,24)(19,25)(20,21)(23,28)(26,27);(1,2,3,4)(5,7,9,8)(6,10,11,12)(13,15)(14,19)(16,23)(17,22)(18,25)(20,27)(21,26)(24,28);(5,8,9,7)(6,10,11,12)(13,15)(14,19)(16,23)(17,22)(18,25)(20,27)(21,26)(24,28);(15,22)(19,26)(23,28)(25,27)Polycyclic presentation:
2^9|a^2=i,c^2=g,d^2=h,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,e^d=ei86 conjugacy classes — view the full character table.
| maximal subgroup of | #72249598, #72249615, #72249662 |
| maximal normal subgroup of | #72249598, #72249615, #72249662 |
| maximal quotient of | #72264661, #72264780, #72308637, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6370079);