Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2601293 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3866 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,9)(10,12)(13,14,18,16)(15,20,17,19)(22,25)(26,28);(5,7)(6,10)(8,11)(9,12)(13,15,18,17)(14,19,16,20)(23,27)(26,28);(13,14,18,16)(15,20,17,19)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,8,9,6)(7,11,12,10)(13,16,18,14)(15,20,17,19)(21,23)(22,26)(24,27)(25,28);(5,7)(6,11)(8,10)(9,12)(13,17,18,15)(14,19,16,20)(21,22)(23,26)(24,25)(27,28)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=h,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef74 conjugacy classes — view the full character table.
| maximal subgroup of | #72263509, #72263528, #72263543, … |
| maximal normal subgroup of | #72263509, #72263528, #72263543, … |
| maximal quotient of | #72293975, C2×#2615702, #72333050 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2522898);